First slide

Evaluation of definite integrals

Question

$\int_{0}^{a} {(a^{2} - x^{2})}^{5 / 2} d x =$

Moderate

A

$\frac{2 π a^{6}}{32}$
B

$\frac{5 π a^{6}}{32}$
C

$\frac{5 π a^{6}}{30}$
D

$\frac{3 π a^{6}}{35}$

Solution

$\begin{array}{l} Put x = a \sin θ \Rightarrow d x = a \cos θ d θ \\ G.I = \int_{0}^{π / 2} {(a^{2} - a^{2} \sin^{2} θ)}^{5 / 2} (a \cos θ) d θ = a^{6} \int_{0}^{π / 2} \cos^{5} θ \cos θ d θ \\ = a^{6} \int_{0}^{π / 2} \cos^{6} θ d θ \\ = a^{6} \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{π}{2} = \frac{5 π a^{6}}{32} \end{array}$

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