Question

Evaluate the integral
${\int }_0^{{\pi }_{/}2}\frac{cosx}{1+si{n}^{2}x}dx$

• A. $\pi$
• B. $\pi /3$
• C. $\pi /2$
• D. $\pi /4$

Answer 1

Answer: (D)

$\pi /4$

${\int }_0^{\frac{\pi }{2}}\frac{cosx}{1+si{n}^{2}x}dx={\int }_0^{\frac{\pi }{2}}\frac{dsinx}{1+si{n}^{2}x}$

${\int }_0^{\frac{\pi }{2}}\frac{dsinx}{1+si{n}^{2}x}=\left[\right(ta{n}^{-1}\left(sinx\right)+c){]}_0^{\frac{\pi }{2}}(\int \frac{1}{1+x^2}={\tan }^{-1}x)$

$=\left(ta{n}^{-1}\right(sin\left(\frac{\pi }{2}\right)+c)-\left(ta{n}^{-1}\right(sin0)+c)$

$=ta{n}^{-1}\left(1\right)$

$=\frac{\pi }{4}$