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
\(2\)
\(\dfrac{3}{4}\)
\(0\)
\(-2\)