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
\(\cos x + x\sin x\)
\(x\sin x\)
\(x\cos x\)
\(\sin x + x\cos x\)