Q115 / 160 · TS EAMCET 2007 · Mathematics

$\int_0^{2 \pi} \sin ^6 x \cos ^5 x d x$ is equal to

  • A
    $2 \pi$
  • B
    $\pi / 2$
  • Correct answer
    C
    $0$
  • D
    $-\pi$

Solution

Let $\begin{aligned} I & =\int_0^{2 \pi} \sin ^6 x \cos ^5 x d x \\ I & =2 \int_0^\pi \sin ^6 x \cos ^5 x d x\end{aligned}$ $[\because f(2 \pi-x)=f(x)]$ Let $f(x)=\sin ^6 x \cos ^5 x$ $[\because f(\pi-x)=-f(x)]$ $\begin{aligned} f(\pi-x) & =\sin ^6(\pi-x) \cdot \cos ^5(\pi-x) \\ & =\sin ^6 x \cdot\left(-\cos ^5 x\right) \\ & =-\sin ^6 x \cdot \cos ^5 x \\ & =-f(x) \\ \therefore \quad I & =0\end{aligned}$

Hence, the correct answer is option: 'C'.

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