Q116 / 160 · TS EAMCET 2024 (10 May Shift 2) · Mathematics

$\frac{3}{25} \int_0^{25 \pi} \sqrt{\left|\cos x-\cos ^3 x\right|} d x=$

  • A
    -8
  • Correct answer
    B
    4
  • C
    1
  • D
    0

Solution

$\mathrm{I}=\frac{3}{25} \int_0^{25 \pi} \sqrt{\left|\cos x-\cos ^3 x\right|} d x$[Period = $\pi$] $=\frac{3}{25} \times 25 \int_0^\pi \sqrt{\left|\cos x-\cos ^3 x\right|} d x$ $\begin{aligned} & =3 \int_0^{\frac{\pi}{2}} \sqrt{\cos x-\cos ^3 x} d x+3 \int_{\frac{\pi}{2}}^\pi \sqrt{\cos ^3 x-\cos x} d x \\ & =3 \int_0^{\frac{\pi}{2}} \sqrt{\cos x} \sin x d x+3 \int_0^{\frac{\pi}{2}} \sqrt{-\cos x} \sin x d x\end{aligned}$ $=3\left[\frac{-(\cos x)^{\frac{3}{2}}}{\frac{3}{2}}\right]_0^{\frac{\pi}{2}}+3\left[\frac{(-\cos x)^{\frac{3}{2}}}{\frac{3}{2}}\right]_{\frac{\pi}{2}}^\pi=4$

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

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