Find an anti derivative (or integral) of \(\sin 2x\) by the method of inspection.
\(-\frac{1}{2}\cos 2x\)
Find an anti derivative (or integral) of \(\cos 3x\) by the method of inspection.
\(\frac{1}{3}\sin 3x\)
Find an anti derivative (or integral) of \(e^{2x}\) by the method of inspection.
\(\frac{1}{2}e^{2x}\)
Find an anti derivative (or integral) of \((ax + b)^2\) by the method of inspection.
\(\frac{1}{3a}(ax + b)^3\)
Find an anti derivative (or integral) of \(\sin 2x - 4e^{3x}\) by the method of inspection.
\(-\frac{1}{2}\cos 2x - \frac{4}{3}e^{3x}\)
Evaluate the integral \(\int (4e^{3x} + 1)\,dx\).
\(\frac{4}{3}e^{3x} + x + C\)
Evaluate the integral \(\int x^2\left(1 - \frac{1}{x^2}\right)dx\).
\(\frac{x^3}{3} - x + C\)
Evaluate the integral \(\int (ax^2 + bx + c)\,dx\).
\(\frac{ax^3}{3} + \frac{bx^2}{2} + cx + C\)
Evaluate the integral \(\int (2x^2 + e^x)\,dx\).
\(\frac{2}{3}x^3 + e^x + C\)
Evaluate the integral \(\int \left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)^2 dx\).
\(\frac{x^2}{2} + \log|x| - 2x + C\)
Evaluate the integral \(\int \frac{x^3 + 5x^2 - 4}{x^2} dx\).
\(\frac{x^2}{2} + 5x + \frac{4}{x} + C\)
Evaluate the integral \(\int \frac{x^3 + 3x + 4}{\sqrt{x}} dx\).
\(\frac{2}{7}x^{7/2} + 2x^{3/2} + 8\sqrt{x} + C\)
Evaluate the integral \(\int \frac{x^3 - x^2 + x - 1}{x - 1} dx\).
\(\frac{x^3}{3} + x + C\)
Evaluate the integral \(\int (1 - x)\sqrt{x}\,dx\).
\(\frac{2}{3}x^{3/2} - \frac{2}{5}x^{5/2} + C\)
Evaluate the integral \(\int \sqrt{x}(3x^2 + 2x + 3)\,dx\).
\(\frac{6}{7}x^{7/2} + \frac{4}{5}x^{5/2} + 2x^{3/2} + C\)
Evaluate the integral \(\int (2x - 3\cos x + e^x)\,dx\).
\(x^2 - 3\sin x + e^x + C\)
Evaluate the integral \(\int (2x^2 - 3\sin x + 5\sqrt{x})\,dx\).
\(\frac{2}{3}x^3 + 3\cos x + \frac{10}{3}x^{3/2} + C\)
Evaluate the integral \(\int \sec x(\sec x + \tan x)\,dx\).
\(\tan x + \sec x + C\)
Evaluate the integral \(\int \frac{\sec^2 x}{\cosec^2 x}\,dx\).
\(\tan x - x + C\)
Evaluate the integral \(\int \frac{2 - 3\sin x}{\cos^2 x}\,dx\).
\(2\tan x - 3\sec x + C\)
The anti derivative of \(\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)\) equals
\(\frac{1}{3}x^{1/3} + 2x^{1/2} + C\)
\(\frac{2}{3}x^{2/3} + \frac{1}{2}x^2 + C\)
\(\frac{2}{3}x^{3/2} + 2x^{1/2} + C\)
\(\frac{3}{2}x^{3/2} + \frac{1}{2}x^{1/2} + C\)
If \(\frac{d}{dx}f(x) = 4x^3 - \frac{3}{x^4}\) such that \(f(2) = 0\), then \(f(x)\) is
\(x^4 + \frac{1}{x^3} - \frac{129}{8}\)
\(x^3 + \frac{1}{x^4} + \frac{129}{8}\)
\(x^4 + \frac{1}{x^3} + \frac{129}{8}\)
\(x^3 + \frac{1}{x^4} - \frac{129}{8}\)
Evaluate the integral \(\int \frac{2x}{1+x^2}\,dx\).
\(\log(1+x^2) + C\)
Evaluate the integral \(\int \frac{(\log x)^2}{x}\,dx\).
\(\frac{1}{3}(\log|x|)^3 + C\)
Evaluate the integral \(\int \frac{1}{x + x\log x}\,dx\).
\(\log|1+\log x| + C\)
Evaluate the integral \(\int \sin x\,\sin(\cos x)\,dx\).
\(\cos(\cos x) + C\)
Evaluate the integral \(\int \sin(ax+b)\,\cos(ax+b)\,dx\).
\(-\frac{1}{4a}\cos\big(2(ax+b)\big) + C\)
Evaluate the integral \(\int \sqrt{ax+b}\,dx\).
\(\frac{2}{3a}(ax+b)^{3/2} + C\)
Evaluate the integral \(\int x\sqrt{x+2}\,dx\).
\(\frac{2}{5}(x+2)^{5/2} - \frac{4}{3}(x+2)^{3/2} + C\)
Evaluate the integral \(\int x\sqrt{1+2x^2}\,dx\).
\(\frac{1}{6}(1+2x^2)^{3/2} + C\)
Evaluate the integral \(\int (4x+2)\sqrt{x^2+x+1}\,dx\).
\(\frac{4}{3}(x^2+x+1)^{3/2} + C\)
Evaluate the integral \(\int \frac{1}{x-\sqrt{x}}\,dx\).
\(2\log|\sqrt{x}-1| + C\)
Evaluate the integral \(\int \frac{x}{\sqrt{x+4}}\,dx\), \(x>0\).
\(\frac{2}{3}\sqrt{x+4}(x-8) + C\)
Evaluate the integral \(\int (x^3-1)^{1/3}x^5\,dx\).
\(\frac{1}{7}(x^3-1)^{7/3} + \frac{1}{4}(x^3-1)^{4/3} + C\)
Evaluate the integral \(\int \frac{x^2}{(2+3x^3)^3}\,dx\).
\(-\frac{1}{18(2+3x^3)^2} + C\)
Evaluate the integral \(\int \frac{1}{x(\log x)^m}\,dx\), \(x>0\), \(m\ne 1\).
\(\frac{(\log x)^{1-m}}{1-m} + C\)
Evaluate the integral \(\int \frac{x}{9-4x^2}\,dx\).
\(-\frac{1}{8}\log|9-4x^2| + C\)
Evaluate the integral \(\int e^{2x+3}\,dx\).
\(\frac{1}{2}e^{2x+3} + C\)
Evaluate the integral \(\int \frac{x}{e^{x^2}}\,dx\).
\(-\frac{1}{2e^{x^2}} + C\)
Evaluate the integral \(\int \frac{e^{\tan^{-1}x}}{1+x^2}\,dx\).
\(e^{\tan^{-1}x} + C\)
Evaluate the integral \(\int \frac{e^{2x}-1}{e^{2x}+1}\,dx\).
\(\log(e^x + e^{-x}) + C\)
Evaluate the integral \(\int \frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}\,dx\).
\(\frac{1}{2}\log(e^{2x}+e^{-2x}) + C\)
Evaluate the integral \(\int \tan^2(2x-3)\,dx\).
\(\frac{1}{2}\tan(2x-3) - x + C\)
Evaluate the integral \(\int \sec^2(7-4x)\,dx\).
\(-\frac{1}{4}\tan(7-4x) + C\)
Evaluate the integral \(\int \frac{\sin^{-1}x}{\sqrt{1-x^2}}\,dx\).
\(\frac{1}{2}(\sin^{-1}x)^2 + C\)
Evaluate the integral \(\int \frac{2\cos x - 3\sin x}{6\cos x + 4\sin x}\,dx\).
\(\frac{1}{2}\log|2\sin x + 3\cos x| + C\)
Evaluate the integral \(\int \frac{1}{\cos^2 x\,(1-\tan x)^2}\,dx\).
\(\frac{1}{1-\tan x} + C\)
Evaluate the integral \(\int \frac{\cos\sqrt{x}}{\sqrt{x}}\,dx\).
\(2\sin\sqrt{x} + C\)
Evaluate the integral \(\int \sqrt{\sin 2x}\,\cos 2x\,dx\).
\(\frac{1}{3}(\sin 2x)^{3/2} + C\)
Evaluate the integral \(\int \frac{\cos x}{\sqrt{1+\sin x}}\,dx\).
\(2\sqrt{1+\sin x} + C\)
Evaluate the integral \(\int \cot x\,\log(\sin x)\,dx\).
\(\frac{1}{2}(\log \sin x)^2 + C\)
Evaluate the integral \(\int \frac{\sin x}{1+\cos x}\,dx\).
\(-\log|1+\cos x| + C\)
Evaluate the integral \(\int \frac{\sin x}{(1+\cos x)^2}\,dx\).
\(\frac{1}{1+\cos x} + C\)
Evaluate the integral \(\int \frac{1}{1+\cot x}\,dx\).
\(\frac{x}{2} - \frac{1}{2}\log|\cos x + \sin x| + C\)
Evaluate the integral \(\int \frac{1}{1-\tan x}\,dx\).
\(\frac{x}{2} - \frac{1}{2}\log|\cos x - \sin x| + C\)
Evaluate the integral \(\int \frac{\sqrt{\tan x}}{\sin x\cos x}\,dx\).
\(2\sqrt{\tan x} + C\)
Evaluate the integral \(\int \frac{(1+\log x)^2}{x}\,dx\).
\(\frac{1}{3}(1+\log x)^3 + C\)
Evaluate the integral \(\int \frac{(x+1)(x+\log x)^2}{x}\,dx\).
\(\frac{1}{3}(x+\log x)^3 + C\)
Evaluate the integral \(\int \frac{x^3\sin(\tan^{-1}x^4)}{1+x^8}\,dx\).
\(-\frac{1}{4}\cos(\tan^{-1}x^4) + C\)
\(\int \frac{10x^9 + 10^x\log_e 10}{x^{10}+10^x}\,dx\) equals
\(10^x - x^{10} + C\)
\(10^x + x^{10} + C\)
\((10^x - x^{10})^{-1} + C\)
\(\log(10^x + x^{10}) + C\)
\(\int \frac{dx}{\sin^2 x\cos^2 x}\) equals
\(\tan x + \cot x + C\)
\(\tan x - \cot x + C\)
\(\tan x\cot x + C\)
\(\tan x - \cot 2x + C\)
Evaluate the integral \(\int \sin^2(2x+5)\,dx\).
\(\frac{x}{2}-\frac{1}{8}\sin(4x+10)+C\)
Evaluate the integral \(\int \sin 3x\,\cos 4x\,dx\).
\(-\frac{1}{14}\cos 7x+\frac{1}{2}\cos x+C\)
Evaluate the integral \(\int \cos 2x\,\cos 4x\,\cos 6x\,dx\).
\(\frac{1}{4}\left[\frac{1}{12}\sin 12x + x + \frac{1}{8}\sin 8x + \frac{1}{4}\sin 4x\right]+C\)
Evaluate the integral \(\int \sin^3(2x+1)\,dx\).
\(-\frac{1}{2}\cos(2x+1)+\frac{1}{6}\cos^3(2x+1)+C\)
Evaluate the integral \(\int \sin^3 x\,\cos^3 x\,dx\).
\(\frac{1}{6}\cos^6 x-\frac{1}{4}\cos^4 x+C\)
Evaluate the integral \(\int \sin x\,\sin 2x\,\sin 3x\,dx\).
\(\frac{1}{4}\left[\frac{1}{6}\cos 6x-\frac{1}{4}\cos 4x-\frac{1}{2}\cos 2x\right]+C\)
Evaluate the integral \(\int \sin 4x\,\sin 8x\,dx\).
\(-\frac{1}{2}\left[\frac{1}{4}\sin 4x-\frac{1}{12}\sin 12x\right]+C\)
Evaluate the integral \(\int \frac{1-\cos x}{1+\cos x}\,dx\).
\(2\tan\frac{x}{2}-x+C\)
Evaluate the integral \(\int \frac{\cos x}{1+\cos x}\,dx\).
\(x-\tan\frac{x}{2}+C\)
Evaluate the integral \(\int \sin^4 x\,dx\).
\(\frac{3x}{8}-\frac{1}{4}\sin 2x+\frac{1}{32}\sin 4x+C\)
Evaluate the integral \(\int \cos^4 2x\,dx\).
\(\frac{3x}{8}+\frac{1}{8}\sin 4x+\frac{1}{64}\sin 8x+C\)
Evaluate the integral \(\int \frac{\sin^2 x}{1+\cos x}\,dx\).
\(x-\sin x+C\)
Evaluate the integral \(\int \frac{\cos 2x-\cos 2\alpha}{\cos x-\cos \alpha}\,dx\).
\(2(\sin x + x\cos\alpha)+C\)
Evaluate the integral \(\int \frac{\cos x-\sin x}{1+\sin 2x}\,dx\).
\(-\frac{1}{\cos x+\sin x}+C\)
Evaluate the integral \(\int \tan^3 2x\,\sec 2x\,dx\).
\(\frac{1}{6}\sec^3 2x-\frac{1}{2}\sec 2x+C\)
Evaluate the integral \(\int \tan^4 x\,dx\).
\(\frac{1}{3}\tan^3 x-\tan x+x+C\)
Evaluate the integral \(\int \frac{\sin^3 x+\cos^3 x}{\sin^2 x\cos^2 x}\,dx\).
\(\sec x-\cosec x+C\)
Evaluate the integral \(\int \frac{\cos 2x+2\sin^2 x}{\cos^2 x}\,dx\).
\(\tan x+C\)
Evaluate the integral \(\int \frac{1}{\sin x\cos^3 x}\,dx\).
\(\log|\tan x|+\frac{1}{2}\tan^2 x+C\)
Evaluate the integral \(\int \frac{\cos 2x}{(\cos x+\sin x)^2}\,dx\).
\(\log|\cos x+\sin x|+C\)
Evaluate the integral \(\int \sin^{-1}(\cos x)\,dx\).
\(\frac{\pi x}{2}-\frac{x^2}{2}+C\)
Evaluate the integral \(\int \frac{1}{\cos(x-a)\,\cos(x-b)}\,dx\).
\(\frac{1}{\sin(a-b)}\log\left|\frac{\cos(x-a)}{\cos(x-b)}\right|+C\)
\(\int \frac{\sin^2 x-\cos^2 x}{\sin^2 x\cos^2 x}\,dx\) is equal to
\(\tan x+\cot x+C\)
\(\tan x+\cosec x+C\)
\(-\tan x+\cot x+C\)
\(\tan x+\sec x+C\)
\(\int \frac{e^x(1+x)}{\cos^2(e^x x)}\,dx\) equals
\(-\cot(e^x x)+C\)
\(\tan(xe^x)+C\)
\(\tan(e^x)+C\)
\(\cot(e^x)+C\)
Integrate: \(\dfrac{3x^2}{x^6+1}\)
\(\tan^{-1}(x^3)+C\)
Integrate: \(\dfrac{1}{\sqrt{1+4x^2}}\)
\(\dfrac{1}{2}\log\left|2x+\sqrt{1+4x^2}\right|+C\)
Integrate: \(\dfrac{1}{\sqrt{(2-x)^2+1}}\)
\(\log\left|\dfrac{1}{2-x+\sqrt{x^2-4x+5}}\right|+C\)
Integrate: \(\dfrac{1}{\sqrt{9-25x^2}}\)
\(\dfrac{1}{5}\sin^{-1}\left(\dfrac{5x}{3}\right)+C\)
Integrate: \(\dfrac{3x}{1+2x^4}\)
\(\dfrac{3}{2\sqrt{2}}\tan^{-1}(\sqrt{2}\,x^2)+C\)
Integrate: \(\dfrac{x^2}{1-x^6}\)
\(\dfrac{1}{6}\log\left|\dfrac{1+x^3}{1-x^3}\right|+C\)
Integrate: \(\dfrac{x-1}{\sqrt{x^2-1}}\)
\(\sqrt{x^2-1}-\log\left|x+\sqrt{x^2-1}\right|+C\)
Integrate: \(\dfrac{x^2}{\sqrt{x^6+a^6}}\)
\(\dfrac{1}{3}\log\left|x^3+\sqrt{x^6+a^6}\right|+C\)
Integrate: \(\dfrac{\sec^2 x}{\sqrt{\tan^2 x+4}}\)
\(\log\left|\tan x+\sqrt{\tan^2 x+4}\right|+C\)
Integrate: \(\dfrac{1}{\sqrt{x^2+2x+2}}\)
\(\log\left|x+1+\sqrt{x^2+2x+2}\right|+C\)
Integrate: \(\dfrac{1}{9x^2+6x+5}\)
\(\dfrac{1}{6}\tan^{-1}\left(\dfrac{3x+1}{2}\right)+C\)
Integrate: \(\dfrac{1}{\sqrt{7-6x-x^2}}\)
\(\sin^{-1}\left(\dfrac{x+3}{4}\right)+C\)
Integrate: \(\dfrac{1}{\sqrt{(x-1)(x-2)}}\)
\(\log\left|x-\dfrac{3}{2}+\sqrt{x^2-3x+2}\right|+C\)
Integrate: \(\dfrac{1}{\sqrt{8+3x-x^2}}\)
\(\sin^{-1}\left(\dfrac{2x-3}{\sqrt{41}}\right)+C\)
Integrate: \(\dfrac{1}{\sqrt{(x-a)(x-b)}}\)
\(\log\left|x-\dfrac{a+b}{2}+\sqrt{(x-a)(x-b)}\right|+C\)
Integrate: \(\dfrac{4x+1}{\sqrt{2x^2+x-3}}\)
\(2\sqrt{2x^2+x-3}+C\)
Integrate: \(\dfrac{x+2}{\sqrt{x^2-1}}\)
\(\sqrt{x^2-1}+2\log\left|x+\sqrt{x^2-1}\right|+C\)
Integrate: \(\dfrac{5x-2}{1+2x+3x^2}\)
\(\dfrac{5}{6}\log\left|3x^2+2x+1\right|-\dfrac{11}{3\sqrt{2}}\tan^{-1}\left(\dfrac{3x+1}{\sqrt{2}}\right)+C\)
Integrate: \(\dfrac{6x+7}{\sqrt{(x-5)(x-4)}}\)
\(6\sqrt{x^2-9x+20}+34\log\left|x-\dfrac{9}{2}+\sqrt{x^2-9x+20}\right|+C\)
Integrate: \(\dfrac{x+2}{\sqrt{4x-x^2}}\)
\(-\sqrt{4x-x^2}+4\sin^{-1}\left(\dfrac{x-2}{2}\right)+C\)
Integrate: \(\dfrac{x+2}{\sqrt{x^2+2x+3}}\)
\(\sqrt{x^2+2x+3}+\log\left|x+1+\sqrt{x^2+2x+3}\right|+C\)
Integrate: \(\dfrac{x+3}{x^2-2x-5}\)
\(\dfrac{1}{2}\log\left|x^2-2x-5\right|+\dfrac{2}{\sqrt{6}}\log\left|\dfrac{x-1-\sqrt{6}}{x-1+\sqrt{6}}\right|+C\)
Integrate: \(\dfrac{5x+3}{\sqrt{x^2+4x+10}}\)
\(5\sqrt{x^2+4x+10}-7\log\left|x+2+\sqrt{x^2+4x+10}\right|+C\)
Integrate the rational function \(\dfrac{x}{(x+1)(x+2)}\).
\(\log\left|\dfrac{(x+2)^2}{x+1}\right|+C\)
Integrate the rational function \(\dfrac{1}{x^2-9}\).
\(\dfrac{1}{6}\log\left|\dfrac{x-3}{x+3}\right|+C\)
Integrate the rational function \(\dfrac{3x-1}{(x-1)(x-2)(x-3)}\).
\(\log|x-1|-5\log|x-2|+4\log|x-3|+C\)
Integrate the rational function \(\dfrac{x}{(x-1)(x-2)(x-3)}\).
\(\dfrac{1}{2}\log|x-1|-2\log|x-2|+\dfrac{3}{2}\log|x-3|+C\)
Integrate the rational function \(\dfrac{2x}{x^2+3x+2}\).
\(4\log|x+2|-2\log|x+1|+C\)
Integrate the rational function \(\dfrac{1-x^2}{x(1-2x)}\).
\(\dfrac{x}{2}+\log|x|-\dfrac{3}{4}\log|1-2x|+C\)
Integrate the rational function \(\dfrac{x}{(x^2+1)(x-1)}\).
\(\dfrac{1}{2}\log|x-1|-\dfrac{1}{4}\log(x^2+1)+\dfrac{1}{2}\tan^{-1}x+C\)
Integrate the rational function \(\dfrac{x}{(x-1)^2(x+2)}\).
\(\dfrac{2}{9}\log\left|\dfrac{x-1}{x+2}\right|-\dfrac{1}{3(x-1)}+C\)
Integrate the rational function \(\dfrac{3x+5}{x^3-x^2-x+1}\).
\(\dfrac{1}{2}\log\left|\dfrac{x+1}{x-1}\right|-\dfrac{4}{x-1}+C\)
Integrate the rational function \(\dfrac{2x-3}{(x^2-1)(2x+3)}\).
\(\dfrac{5}{2}\log|x+1|-\dfrac{1}{10}\log|x-1|-\dfrac{12}{5}\log|2x+3|+C\)
Integrate the rational function \(\dfrac{5x}{(x+1)(x^2-4)}\).
\(\dfrac{5}{3}\log|x+1|-\dfrac{5}{2}\log|x+2|+\dfrac{5}{6}\log|x-2|+C\)
Integrate the rational function \(\dfrac{x^3+x+1}{x^2-1}\).
\(\dfrac{x^2}{2}+\dfrac{1}{2}\log|x+1|+\dfrac{3}{2}\log|x-1|+C\)
Integrate the rational function \(\dfrac{2}{(1-x)(1+x^2)}\).
\(-\log|x-1|+\dfrac{1}{2}\log(1+x^2)+\tan^{-1}x+C\)
Integrate the rational function \(\dfrac{3x-1}{(x+2)^2}\).
\(3\log|x+2|+\dfrac{7}{x+2}+C\)
Integrate the rational function \(\dfrac{1}{x^4-1}\).
\(\dfrac{1}{4}\log\left|\dfrac{x-1}{x+1}\right|-\dfrac{1}{2}\tan^{-1}x+C\)
Integrate the rational function \(\dfrac{1}{x(x^n+1)}\).
Hint: multiply numerator and denominator by \(x^{n-1}\) and put \(x^n=t\).
\(\dfrac{1}{n}\log\left|\dfrac{x^n}{x^n+1}\right|+C\)
Integrate \(\dfrac{\cos x}{(1-\sin x)(2-\sin x)}\).
Hint: Put \(\sin x=t\).
\(\log\left|\dfrac{2-\sin x}{1-\sin x}\right|+C\)
Integrate the rational function \(\dfrac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}\).
\(x+\dfrac{2}{\sqrt{3}}\tan^{-1}\left(\dfrac{x}{\sqrt{3}}\right)-3\tan^{-1}\left(\dfrac{x}{2}\right)+C\)
Integrate the rational function \(\dfrac{2x}{(x^2+1)(x^2+3)}\).
\(\dfrac{1}{2}\log\left(\dfrac{x^2+1}{x^2+3}\right)+C\)
Integrate the rational function \(\dfrac{1}{x(x^4-1)}\).
\(\dfrac{1}{4}\log\left|\dfrac{x^4-1}{x^4}\right|+C\)
Integrate \(\dfrac{1}{e^x-1}\).
Hint: Put \(e^x=t\).
\(\log\left(\dfrac{e^x-1}{e^x}\right)+C\)
Choose the correct answer:
\(\displaystyle \int \dfrac{x\,dx}{(x-1)(x-2)}\) equals
(A) \(\log\left|\dfrac{(x-1)^2}{x-2}\right|+C\)
(B) \(\log\left|\dfrac{(x-2)^2}{x-1}\right|+C\)
(C) \(\log\left|\left(\dfrac{x-1}{x-2}\right)^2\right|+C\)
(D) \(\log|(x-1)(x-2)|+C\)
B
Choose the correct answer:
\(\displaystyle \int \dfrac{dx}{x(x^2+1)}\) equals
(A) \(\log|x|-\dfrac{1}{2}\log(x^2+1)+C\)
(B) \(\log|x|+\dfrac{1}{2}\log(x^2+1)+C\)
(C) \(-\log|x|+\dfrac{1}{2}\log(x^2+1)+C\)
(D) \(\dfrac{1}{2}\log|x|+\log(x^2+1)+C\)
A
Find the integral of \(x\sin x\).
\(-x\cos x+\sin x+C\)
Find the integral of \(x\sin 3x\).
\(-\dfrac{x}{3}\cos 3x+\dfrac{1}{9}\sin 3x+C\)
Find the integral of \(x^2 e^x\).
\(e^x(x^2-2x+2)+C\)
Find the integral of \(x\log x\).
\(\dfrac{x^2}{2}\log x-\dfrac{x^2}{4}+C\)
Find the integral of \(x\log 2x\).
\(\dfrac{x^2}{2}\log 2x-\dfrac{x^2}{4}+C\)
Find the integral of \(x^2\log x\).
\(\dfrac{x^3}{3}\log x-\dfrac{x^3}{9}+C\)
Find the integral of \(x\sin^{-1}x\).
\(\dfrac{1}{4}(2x^2-1)\sin^{-1}x+\dfrac{x\sqrt{1-x^2}}{4}+C\)
Find the integral of \(x\tan^{-1}x\).
\(\dfrac{x^2}{2}\tan^{-1}x-\dfrac{x}{2}+\dfrac{1}{2}\tan^{-1}x+C\)
Find the integral of \(x\cos^{-1}x\).
\(\dfrac{(2x^2-1)\cos^{-1}x}{4}-\dfrac{x}{4}\sqrt{1-x^2}+C\)
Find the integral of \((\sin^{-1}x)^2\).
\(x(\sin^{-1}x)^2+2\sqrt{1-x^2}\sin^{-1}x-2x+C\)
Find the integral of \(\dfrac{x\cos^{-1}x}{\sqrt{1-x^2}}\).
\(-\sqrt{1-x^2}\cos^{-1}x+x+C\)
Find the integral of \(x\sec^2 x\).
\(x\tan x+\log|\cos x|+C\)
Find the integral of \(\tan^{-1}x\).
\(x\tan^{-1}x-\dfrac{1}{2}\log(1+x^2)+C\)
Find the integral of \(x(\log x)^2\).
\(\dfrac{x^2}{2}(\log x)^2-\dfrac{x^2}{2}\log x+\dfrac{x^2}{4}+C\)
Find the integral of \((x^2+1)\log x\).
\(\left(\dfrac{x^3}{3}+x\right)\log x-\dfrac{x^3}{9}-x+C\)
Find the integral of \(e^x(\sin x+\cos x)\).
\(e^x\sin x+C\)
Find the integral of \(\dfrac{x e^x}{(1+x)^2}\).
\(\dfrac{e^x}{1+x}+C\)
Find the integral of \(e^x\left(\dfrac{1+\sin x}{1+\cos x}\right)\).
\(e^x\tan\dfrac{x}{2}+C\)
Find the integral of \(e^x\left(\dfrac{1}{x}-\dfrac{1}{x^2}\right)\).
\(\dfrac{e^x}{x}+C\)
Find the integral of \(\dfrac{(x-3)e^x}{(x-1)^3}\).
\(\dfrac{e^x}{(x-1)^2}+C\)
Find the integral of \(e^{2x}\sin x\).
\(\dfrac{e^{2x}}{5}(2\sin x-\cos x)+C\)
Find the integral of \(\sin^{-1}\left(\dfrac{2x}{1+x^2}\right)\).
\(2x\tan^{-1}x-\log(1+x^2)+C\)
\(\displaystyle \int x^2 e^{x^3}\,dx\) equals
\(\dfrac{1}{3}e^{x^3}+C\)
\(\dfrac{1}{3}e^{x^2}+C\)
\(\dfrac{1}{2}e^{x^3}+C\)
\(\dfrac{1}{2}e^{x^2}+C\)
\(\displaystyle \int e^x\sec x(1+\tan x)\,dx\) equals
\(e^x\cos x+C\)
\(e^x\sec x+C\)
\(e^x\sin x+C\)
\(e^x\tan x+C\)
Integrate \(\sqrt{4-x^2}\).
\(\dfrac{1}{2}x\sqrt{4-x^2}+2\sin^{-1}\!\left(\dfrac{x}{2}\right)+C\)
Integrate \(\sqrt{1-4x^2}\).
\(\dfrac{1}{4}\sin^{-1}(2x)+\dfrac{1}{2}x\sqrt{1-4x^2}+C\)
Integrate \(\sqrt{x^2+4x+6}\).
\(\dfrac{x+2}{2}\sqrt{x^2+4x+6}+\log\left|x+2+\sqrt{x^2+4x+6}\right|+C\)
Integrate \(\sqrt{x^2+4x+1}\).
\(\dfrac{x+2}{2}\sqrt{x^2+4x+1}-\dfrac{3}{2}\log\left|x+2+\sqrt{x^2+4x+1}\right|+C\)
Integrate \(\sqrt{1-4x-x^2}\).
\(\dfrac{5}{2}\sin^{-1}\!\left(\dfrac{x+2}{\sqrt{5}}\right)+\dfrac{x+2}{2}\sqrt{1-4x-x^2}+C\)
Integrate \(\sqrt{x^2+4x-5}\).
\(\dfrac{x+2}{2}\sqrt{x^2+4x-5}-\dfrac{9}{2}\log\left|x+2+\sqrt{x^2+4x-5}\right|+C\)
Integrate \(\sqrt{1+3x-x^2}\).
\(\dfrac{2x-3}{4}\sqrt{1+3x-x^2}+\dfrac{13}{8}\sin^{-1}\!\left(\dfrac{2x-3}{\sqrt{13}}\right)+C\)
Integrate \(\sqrt{x^2+3x}\).
\(\dfrac{2x+3}{4}\sqrt{x^2+3x}-\dfrac{9}{8}\log\left|x+\dfrac{3}{2}+\sqrt{x^2+3x}\right|+C\)
Integrate \(\sqrt{1+\dfrac{x^2}{9}}\).
\(\dfrac{x}{6}\sqrt{x^2+9}+\dfrac{3}{2}\log\left|x+\sqrt{x^2+9}\right|+C\)
\(\int \sqrt{1+x^2}\,dx\) is equal to
\(\dfrac{x}{2}\sqrt{1+x^2}+\dfrac{1}{2}\log\left|x+\sqrt{1+x^2}\right|+C\)
\(\dfrac{2}{3}(1+x^2)^{3/2}+C\)
\(\dfrac{2}{3}x(1+x^2)^{3/2}+C\)
\(\dfrac{x^2}{2}\sqrt{1+x^2}+\dfrac{1}{2}x^2\log\left|x+\sqrt{1+x^2}\right|+C\)
\(\int \sqrt{x^2-8x+7}\,dx\) is equal to
\(\dfrac{1}{2}(x-4)\sqrt{x^2-8x+7}+9\log\left|x-4+\sqrt{x^2-8x+7}\right|+C\)
\(\dfrac{1}{2}(x+4)\sqrt{x^2-8x+7}+9\log\left|x+4+\sqrt{x^2-8x+7}\right|+C\)
\(\dfrac{1}{2}(x-4)\sqrt{x^2-8x+7}-3\sqrt{2}\,\log\left|x-4+\sqrt{x^2-8x+7}\right|+C\)
\(\dfrac{1}{2}(x-4)\sqrt{x^2-8x+7}-\dfrac{9}{2}\log\left|x-4+\sqrt{x^2-8x+7}\right|+C\)
Evaluate \(\int_{-1}^{1} (x+1)\,dx\).
2
Evaluate \(\int_{2}^{3} \frac{1}{x}\,dx\).
\(\log\frac{3}{2}\)
Evaluate \(\int_{1}^{2} (4x^3-5x^2+6x+9)\,dx\).
\(\frac{64}{3}\)
Evaluate \(\int_{0}^{\pi/4} \sin 2x\,dx\).
\(\frac{1}{2}\)
Evaluate \(\int_{0}^{\pi/2} \cos 2x\,dx\).
0
Evaluate \(\int_{4}^{5} e^x\,dx\).
\(e^4(e-1)\)
Evaluate \(\int_{0}^{\pi/4} \tan x\,dx\).
\(\frac{1}{2}\log 2\)
Evaluate \(\int_{\pi/6}^{\pi/4} \cosec x\,dx\).
\(\log\left(\frac{\sqrt{2}-1}{2-\sqrt{3}}\right)\)
Evaluate \(\int_{0}^{1} \frac{dx}{\sqrt{1-x^2}}\).
\(\frac{\pi}{2}\)
Evaluate \(\int_{0}^{1} \frac{dx}{1+x^2}\).
\(\frac{\pi}{4}\)
Evaluate \(\int_{2}^{3} \frac{dx}{x^2-1}\).
\(\frac{1}{2}\log\frac{3}{2}\)
Evaluate \(\int_{0}^{\pi/2} \cos^2 x\,dx\).
\(\frac{\pi}{4}\)
Evaluate \(\int_{2}^{3} \frac{x\,dx}{x^2+1}\).
\(\frac{1}{2}\log 2\)
Evaluate \(\int_{0}^{1} \frac{2x+3}{5x^2+1}\,dx\).
\(\frac{1}{5}\log 6+\frac{3}{\sqrt{5}}\tan^{-1}\sqrt{5}\)
Evaluate \(\int_{0}^{1} x e^{x^2}\,dx\).
\(\frac{1}{2}(e-1)\)
Evaluate \(\int_{1}^{2} \frac{5x^2}{x^2+4x+3}\,dx\).
\(5-\frac{5}{2}\left(9\log\frac{5}{4}-\log\frac{3}{2}\right)\)
Evaluate \(\int_{0}^{\pi/4} (2\sec^2 x+x^3+2)\,dx\).
\(\frac{\pi^4}{1024}+\frac{\pi}{2}+2\)
Evaluate \(\int_{0}^{\pi} \left(\sin^2\frac{x}{2}-\cos^2\frac{x}{2}\right)\,dx\).
0
Evaluate \(\int_{0}^{2} \frac{6x+3}{x^2+4}\,dx\).
\(3\log 2+\frac{3\pi}{8}\)
Evaluate \(\int_{0}^{1} \left(x e^x+\sin\frac{\pi x}{4}\right)\,dx\).
\(1+\frac{4}{\pi}-\frac{2\sqrt{2}}{\pi}\)
\(\int_{1}^{\sqrt{3}} \frac{dx}{1+x^2}\) equals
\(\frac{\pi}{3}\)
\(\frac{2\pi}{3}\)
\(\frac{\pi}{6}\)
\(\frac{\pi}{12}\)
\(\int_{0}^{2/3} \frac{dx}{4+9x^2}\) equals
\(\frac{\pi}{6}\)
\(\frac{\pi}{12}\)
\(\frac{\pi}{24}\)
\(\frac{\pi}{4}\)
Evaluate the integral \(\int_{0}^{1} \frac{x}{x^2+1}\,dx\).
\(\frac{1}{2}\log 2\)
Evaluate the integral \(\int_{0}^{\pi/2} \sqrt{\sin\phi\,\cos^5\phi}\,d\phi\).
\(\frac{64}{231}\)
Evaluate the integral \(\int_{0}^{1} \sin^{-1}\!\left(\frac{2x}{1+x^2}\right)\,dx\).
\(\frac{\pi}{2}-\log 2\)
Evaluate the integral \(\int_{0}^{2} x\sqrt{x+2}\,dx\) (Put \(x+2=t^2\)).
\(\frac{16\sqrt{2}}{15}(\sqrt{2}+1)\)
Evaluate the integral \(\int_{0}^{\pi/2} \frac{\sin x}{1+\cos^2 x}\,dx\).
\(\frac{\pi}{4}\)
Evaluate the integral \(\int_{0}^{2} \frac{dx}{x+4-x^2}\).
\(\frac{1}{\sqrt{17}}\log\left(\frac{21+5\sqrt{17}}{4}\right)\)
Evaluate the integral \(\int_{-1}^{1} \frac{dx}{x^2+2x+5}\).
\(\frac{\pi}{8}\)
Evaluate the integral \(\int_{1}^{2} \left(\frac{1}{x}-\frac{1}{2x^2}\right)e^{2x}\,dx\).
\(\frac{e^2(e^2-2)}{4}\)
The value of the integral \(\int_{1/3}^{1} \frac{(x-x^3)^{1/3}}{x^4}\,dx\) is
\(6\)
\(0\)
\(3\)
\(4\)
If \(f(x)=\int_{0}^{x} t\sin t\,dt\), then \(f'(x)\) is
\(\cos x + x\sin x\)
\(x\sin x\)
\(x\cos x\)
\(\sin x + x\cos x\)
Evaluate \(\int_{0}^{\pi/2} \cos^{2} x\,dx\).
\(\dfrac{\pi}{4}\)
Evaluate \(\int_{0}^{\pi/2} \dfrac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,dx\).
\(\dfrac{\pi}{4}\)
Evaluate \(\int_{0}^{\pi/2} \dfrac{\sin^{3/2} x}{\sin^{3/2} x+\cos^{3/2} x}\,dx\).
\(\dfrac{\pi}{4}\)
Evaluate \(\int_{0}^{\pi/2} \dfrac{\cos^{5} x}{\sin^{5} x+\cos^{5} x}\,dx\).
\(\dfrac{\pi}{4}\)
Evaluate \(\int_{-5}^{5} |x+2|\,dx\).
\(29\)
Evaluate \(\int_{2}^{8} |x-5|\,dx\).
\(9\)
Evaluate \(\int_{0}^{1} x(1-x)^{n}\,dx\).
\(\dfrac{1}{(n+1)(n+2)}\)
Evaluate \(\int_{0}^{\pi/4} \log(1+\tan x)\,dx\).
\(\dfrac{\pi}{8}\log 2\)
Evaluate \(\int_{0}^{2} x\sqrt{2-x}\,dx\).
\(\dfrac{16\sqrt{2}}{15}\)
Evaluate \(\int_{0}^{\pi/2} \big(2\log(\sin x)-\log(\sin 2x)\big)\,dx\).
\(\dfrac{\pi}{2}\log\left(\dfrac{1}{2}\right)\)
Evaluate \(\int_{-\pi/2}^{\pi/2} \sin^{2} x\,dx\).
\(\dfrac{\pi}{2}\)
Evaluate \(\int_{0}^{\pi} \dfrac{x}{1+\sin x}\,dx\).
\(\pi\)
Evaluate \(\int_{-\pi/2}^{\pi/2} \sin^{7} x\,dx\).
\(0\)
Evaluate \(\int_{0}^{2\pi} \cos^{5} x\,dx\).
\(0\)
Evaluate \(\int_{0}^{\pi/2} \dfrac{\sin x-\cos x}{1+\sin x\cos x}\,dx\).
\(0\)
Evaluate \(\int_{0}^{\pi} \log(1+\cos x)\,dx\).
\(-\pi\log 2\)
Evaluate \(\int_{0}^{a} \dfrac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}\,dx\).
\(\dfrac{a}{2}\)
Evaluate \(\int_{0}^{4} |x-1|\,dx\).
\(5\)
Show that \(\int_{0}^{a} f(x)g(x)\,dx = 2\int_{0}^{a} f(x)\,dx\), if \(f\) and \(g\) are defined as \(f(x)=f(a-x)\) and \(g(x)+g(a-x)=4\).
The value of \(\int_{-\pi/2}^{\pi/2} \big(x^{3}+x\cos x+\tan^{5} x+1\big)\,dx\) is
\(0\)
\(2\)
\(\pi\)
\(1\)
The value of \(\int_{0}^{\pi/2} \log\left(\dfrac{4+3\sin x}{4+3\cos x}\right)\,dx\) is
\(2\)
\(\dfrac{3}{4}\)
\(0\)
\(-2\)
Integrate \( \frac{1}{x - x^3} \).
\( \frac{1}{2} \log \left| \frac{x^2}{1 - x^2} \right| + C \)
Integrate \( \frac{1}{\sqrt{x+a} + \sqrt{x+b}} \).
\( \frac{2}{3(a-b)} \left[ (x+a)^{3/2} - (x+b)^{3/2} \right] + C \)
Integrate \( \frac{1}{x\sqrt{ax - x^2}} \).
\( -\frac{2}{a} \sqrt{\frac{a-x}{x}} + C \)
Integrate \( \frac{1}{x^2 (x^4 + 1)^{3/4}} \).
\( -\left(1 + \frac{1}{x^4}\right)^{1/4} + C \)
Integrate \( \frac{1}{x^{1/2} + x^{1/3}} \).
\( 2\sqrt{x} - 3x^{1/3} + 6x^{1/6} - 6\log(1 + x^{1/6}) + C \)
Integrate \( \frac{5x}{(x+1)(x^2+9)} \).
\( -\frac{1}{2}\log|x+1| + \frac{1}{4}\log(x^2+9) + \frac{3}{2}\tan^{-1}\frac{x}{3} + C \)
Integrate \( \frac{\sin x}{\sin(x-a)} \).
\( \sin a \log|\sin(x-a)| + x\cos a + C \)
Integrate \( \frac{e^{5\log x} - e^{4\log x}}{e^{3\log x} - e^{2\log x}} \).
\( \frac{x^3}{3} + C \)
Integrate \( \frac{\cos x}{\sqrt{4 - \sin^2 x}} \).
\( \sin^{-1}\left( \frac{\sin x}{2} \right) + C \)
Integrate \( \frac{\sin^8 x - \cos^8 x}{1 - 2\sin^2 x\cos^2 x} \).
\( -\frac{1}{2}\sin 2x + C \)
Integrate \( \frac{1}{\cos(x+a)\cos(x+b)} \).
\( \frac{1}{\sin(a-b)} \log\left| \frac{\cos(x+b)}{\cos(x+a)} \right| + C \)
Integrate \( \frac{x^3}{\sqrt{1 - x^8}} \).
\( \frac{1}{4} \sin^{-1}(x^4) + C \)
Integrate \( \frac{e^x}{(1+e^x)(2+e^x)} \).
\( \log\left( \frac{1+e^x}{2+e^x} \right) + C \)
Integrate \( \frac{1}{(x^2+1)(x^2+4)} \).
\( \frac{1}{3}\tan^{-1}x - \frac{1}{6}\tan^{-1}\frac{x}{2} + C \)
Integrate \( \cos^3 x\, e^{\log \sin x} \).
\( -\frac{1}{4}\cos^4 x + C \)
Integrate \( e^{3\log x}(x^4+1)^{-1} \).
\( \frac{1}{4}\log(x^4+1) + C \)
Integrate \( f'(ax+b)[f(ax+b)]^n \).
\( \frac{[f(ax+b)]^{n+1}}{a(n+1)} + C \)
Integrate \( \frac{1}{\sqrt{\sin^3 x\, \sin(x+\alpha)}} \).
\( -\frac{2}{\sin \alpha} \sqrt{ \frac{\sin(x+\alpha)}{\sin x} } + C \)
Integrate \( \sqrt{ \frac{1-\sqrt{x}}{1+\sqrt{x}} } \).
\( -2\sqrt{1-x} + \cos^{-1}\sqrt{x} + \sqrt{x-x^2} + C \)
Integrate \( \frac{2+\sin 2x}{1+\cos 2x} e^x \).
\( e^x \tan x + C \)
Integrate \( \frac{x^2+x+1}{(x+1)^2(x+2)} \).
\( -2\log|x+1| - \frac{1}{x+1} + 3\log|x+2| + C \)
Integrate \( \tan^{-1}\sqrt{ \frac{1-x}{1+x} } \).
\( \frac{1}{2}\left[x\cos^{-1}x - \sqrt{1-x^2}\right] + C \)
Integrate \( \frac{\sqrt{x^2+1}[\log(x^2+1) - 2\log x]}{x^4} \).
\( -\frac{1}{3}\left(1+\frac{1}{x^2}\right)^{3/2}\left[ \log\left(1+\frac{1}{x^2}\right) - \frac{2}{3} \right] + C \)
Evaluate the definite integral:
\( \displaystyle \int_{\pi/2}^{\pi} e^{x}\left(\frac{1-\sin x}{1-\cos x}\right)\,dx \)
\( e^{\pi/2} \)
Evaluate the definite integral:
\( \displaystyle \int_{0}^{\pi/4} \frac{\sin x\,\cos x}{\cos^4 x+\sin^4 x}\,dx \)
\( \frac{\pi}{8} \)
Evaluate the definite integral:
\( \displaystyle \int_{0}^{\pi/2} \frac{\cos^2 x}{\cos^2 x+4\sin^2 x}\,dx \)
\( \frac{\pi}{6} \)
Evaluate the definite integral:
\( \displaystyle \int_{\pi/6}^{\pi/3} \frac{\sin x+\cos x}{\sqrt{\sin 2x}}\,dx \)
\( 2\sin^{-1}\left(\frac{\sqrt{3}-1}{2}\right) \)
Evaluate the definite integral:
\( \displaystyle \int_{0}^{1} \frac{dx}{\sqrt{1+x}-\sqrt{x}} \)
\( \frac{4\sqrt{2}}{3} \)
Evaluate the definite integral:
\( \displaystyle \int_{0}^{\pi/4} \frac{\sin x+\cos x}{9+16\sin 2x}\,dx \)
\( \frac{1}{40}\log 9 \)
Evaluate the definite integral:
\( \displaystyle \int_{0}^{\pi/2} \sin 2x\,\tan^{-1}(\sin x)\,dx \)
\( \frac{\pi}{2}-1 \)
Evaluate the definite integral:
\( \displaystyle \int_{1}^{4} \left(|x-1|+|x-2|+|x-3|\right)\,dx \)
\( \frac{19}{2} \)
Prove that:
\( \displaystyle \int_{1}^{3} \frac{dx}{x^2(x+1)} = \frac{2}{3}+\log\left(\frac{2}{3}\right) \)
Prove that:
\( \displaystyle \int_{0}^{1} x e^{x}\,dx = 1 \)
Prove that:
\( \displaystyle \int_{-1}^{1} x^{17}\cos^{4}x\,dx = 0 \)
Prove that:
\( \displaystyle \int_{0}^{\pi/2} \sin^{3}x\,dx = \frac{2}{3} \)
Prove that:
\( \displaystyle \int_{0}^{\pi/4} 2\tan^{3}x\,dx = 1-\log 2 \)
Prove that:
\( \displaystyle \int_{0}^{1} \sin^{-1}x\,dx = \frac\pi2-1 \)
Evaluate the integral:
\( \displaystyle \int \frac{dx}{e^{x}+e^{-x}} \)
\( \tan^{-1}(e^{x}) + C \)
\( \tan^{-1}(e^{-x}) + C \)
\( \log(e^{x}-e^{-x}) + C \)
\( \log(e^{x}+e^{-x}) + C \)
Evaluate the integral:
\( \displaystyle \int \frac{\cos 2x}{(\sin x+\cos x)^2}\,dx \)
\( \frac{-1}{\sin x+\cos x} + C \)
\( \log |\sin x+\cos x| + C \)
\( \log |\sin x-\cos x| + C \)
\( \frac{1}{(\sin x+\cos x)^2} + C \)
If \( f(a+b-x)=f(x) \), then the value of
\( \displaystyle \int_{a}^{b} x f(x)\,dx \) is equal to
\( \frac{a+b}{2} \int_{a}^{b} f(b-x)\,dx \)
\( \frac{a+b}{2} \int_{a}^{b} f(b+x)\,dx \)
\( \frac{b-a}{2} \int_{a}^{b} f(x)\,dx \)
\( \frac{a+b}{2} \int_{a}^{b} f(x)\,dx \)