Cho hàm số f(x) có \(f\left...

A

- Hướng dẫn giải

Ta có:

\(\begin{array}{l}f\left( x \right) = \smallint \sin x.si{n^2}2xdx\;\\ = 4\smallint \sin x\left( {1 - co{s^2}x} \right)co{s^2}xdx\\ = - 4\smallint \left( {co{s^2}x - co{s^4}x} \right)d\left( {cosx} \right)\\ = \frac{{ - 4}}{3}co{s^3}x + \frac{4}{5}co{s^5}x + C\end{array}\)

\(\begin{array}{l}f\left( {\frac{\pi }{2}} \right) = 0 \Rightarrow C = 0\\ \Rightarrow \int\limits_0^{\frac{\pi }{2}} {f(x)dx} = \frac{{ - 104}}{{225}}\end{array}\)