First slide

Evaluation of definite integrals

Question

The value of the definite integral $\int_{a + 2 π}^{a + 5 π / 2} \{\sin^{- 1} (\cos x) + \cos^{- 1} (\sin x)\} dx$ is equal to

Moderate

A

$\frac{π^{2}}{8}$
B

$\frac{π^{2}}{4}$
C

$\frac{π^{2}}{2}$
D

$π^{2}$

Solution

$∵ \sin^{- 1} (\cos x) + \cos^{- 1} (\sin x)$ is periodic with period $2 π$ then
$\begin{array}{l} \int_{a + 2 π}^{a + 5 π / 2} \{\sin^{- 1} (\cos x) + \cos^{- 1} (\sin x)\} dx \\ = \int_{0}^{π / 2} \{\sin^{- 1} (\cos x) + \cos^{- 1} (\sin x)\} dx \\ = \int_{0}^{π / 2} [\sin^{- 1} \{\cos (\frac{π}{2} - x)\} + \cos^{- 1} \{\sin (\frac{π}{2} - x)\}] dx \\ = \int_{0}^{π / 2} (\sin^{- 1} \sin x + \cos^{- 1} \cos x) dx \\ = 2 \int_{0}^{π / 2} xdx = 2 {\{\frac{x^{2}}{2}\}}_{0}^{\frac{π}{2}} = \frac{π^{2}}{4} \end{array}$

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