First slide

Evaluation of definite integrals

Question

The value $I = \int_{- 1}^{1} (\cos^{- 1} x + \frac{x^{7} - 3 x^{5} + 7 x^{3} - x}{\cos^{2} x}) d x$ is

Moderate

A

$π / 2$
B

$0$
C

$2 π$
D

$π$

Solution

Let $f (x) = \frac{x^{7} - 3 x^{5} + 7 x^{3} - x}{\cos^{2} x}$ then $f (- x) = - f (x)$ so $\int_{- 1}^{1} f (x) d x = 0$ (Property 12)
Let $g (x) = \cos^{- 1} x$ then $I_{1} = \int_{- 1}^{1} \cos^{- 1} x d x$
$= \int_{- 1}^{1} \cos^{- 1} (- x) d x$ (Property 8)
$= \int_{- 1}^{1} (π - \cos^{- 1} x) d x = 2 π - I_{1}$
Thus $I = I_{1} = π .$

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