TS EAMCET · Maths · Definite Integration
\(\int_0^\pi\left(\sin ^3 x+\cos ^2 x\right)^2 d x=\)
- A \(\frac{15 \pi}{16}+\frac{8}{15}\)
- B \(\frac{11 \pi}{16}+\frac{8}{15}\)
- C \(\frac{15 \pi}{16}+\frac{4}{15}\)
- D \(\frac{11 \pi}{16}+\frac{4}{15}\)
Answer & Solution
Correct Answer
(B) \(\frac{11 \pi}{16}+\frac{8}{15}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text {} \quad I=\int_0^\pi\left(\sin ^3 x+\cos ^2 x\right)^2 d x \\ & =\int_0^\pi \sin ^6 x+\cos ^4 x+2 \cos ^2 x \sin ^3 x d x \\ & =2 \int_0^{\frac{\pi}{2}} \sin ^6 x d x+2 \int_0^{\frac{\pi}{2}} \cos ^4 x d x+\int_0^{\frac{\pi}{2}} \cos ^2 x \sin ^3 x d x…
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