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  4. If ∫ limits01 x6 √1-x2 d x=(λ π/k) then find λ +k

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Q. If $\int\limits_{0}^{1} x^{6} \sqrt{1-x^{2}} d x=\frac{\lambda \pi}{k}$ then find $\lambda +k$

Integrals

Solution:

Let $x=\sin \theta \Rightarrow d x=\cos \theta d \theta$
$\int\limits_{0}^{\pi / 2} \sin ^{6} \theta \cdot \cos ^{2} \theta d \theta$
$=\frac{5 \cdot 3 \cdot 1 \cdot 1}{8 \cdot 6 \cdot 4 \cdot 2} \times \frac{\pi}{2}=\frac{5 \pi}{256}$