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
\( \frac{a+b}{2} \int_{a}^{b} f(b-x)\,dx \)
\( \frac{a+b}{2} \int_{a}^{b} f(b+x)\,dx \)
\( \frac{b-a}{2} \int_{a}^{b} f(x)\,dx \)
\( \frac{a+b}{2} \int_{a}^{b} f(x)\,dx \)