First slide

Evaluation of definite integrals

Question

The value of $I = \int_{- π / 2}^{π / 2} \sqrt{\cos x - \cos^{3} x} d x$ is

Moderate

A

0
B

2/3
C

4/3
D

1/3

Solution

$\begin{array}{l} I = \int_{- π / 2}^{π / 2} \sqrt{\cos x} | \sin x | d x \\ = 2 \int_{0}^{π / 2} \sqrt{\cos x} | \sin x | d x \end{array}$
(the integrand is an even function)
$\begin{array}{l} = 2 \int_{0}^{π / 2} \sqrt{\cos x} \sin x d x \\ = - {\frac{4}{3} (\cos x)^{3 / 2}|}_{0}^{π / 2} = \frac{4}{3} . \end{array}$

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