First slide

Evaluation of definite integrals

Question

$\int_{- π / 2}^{π / 2} (x^{14} \sin^{11} x + \sin^{2} x) d x$ is equal to

Easy

A

0
B

$π$
C

$\frac{π}{2}$
D

$- π$

Solution

The function $f (x) = x^{14} \sin^{11} x$ is an odd function,
so $\int_{- π / 2}^{π / 2} x^{14} \sin^{11} x d x = 0$ Thus the given integral is
equal to $\int_{- π / 2}^{π / 2} \sin^{2} x d x = 2 \int_{0}^{π / 2} \sin^{2} x d x = 2 \cdot \frac{1}{2} \cdot \frac{π}{2} = \frac{π}{2}$

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