First slide

Evaluation of definite integrals

Question

$The value of the definite integral \int_{- \frac{1}{2}}^{\frac{1}{2}} (\sin^{- 1} (3 x - 4 x^{3}) - \cos^{- 1} (4 x^{3} - 3 x)) d x =$

Difficult

A

$π$
B

$\frac{π}{2}$
C

$\frac{- π}{2}$
D

$- π$

Solution

$\begin{array}{l} For x \in (- \frac{1}{2}, \frac{1}{2}), \sin^{- 1} (3 x - 4 x^{3}) = 3 \sin^{- 1} x and \\ \cos^{- 1} (4 x^{3} - 3 x) = 2 π - 3 \cos^{- 1} x \end{array}$
$\begin{array}{l} Let I = \int_{- \frac{1}{2}}^{\frac{1}{2}} (\sin^{- 1} (3 x - 4 x^{3}) - \cos^{- 1} (4 x^{3} - 3 x)) d x \\ = \int_{- \frac{1}{2}}^{\frac{1}{2}} (3 \sin^{- 1} x - 2 π + 3 \cos^{- 1} x) d x \\ = \int_{- \frac{1}{2}}^{\frac{1}{2}} (3 (\frac{π}{2}) - 2 π) d x = - \frac{π}{2} (\frac{1}{2} - (- \frac{1}{2})) = - \frac{π}{2} (\sin c e \sin^{- 1} (x) + \cos^{- 1} (x) = \frac{π}{2}) \end{array}$

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