If \textrm{ }\textrm{ } the \textrm{ } value \textrm{ } of \int_{- 1}^{1} \left(sin\right)^{- 1} \sqrt{\left|\right. x \left|\right.} \textrm{ }\textrm{ } \left(cos\right)^{- 1} \left(\right. \frac{x}{1 + x^{2 \left|\right. x \left|\right.}} \left.\right) dx = \frac{\left(aπ\right)^{2}}{b} where a and b are relative prime, then (a + b) is equal to

Engineering

Mathematics

Properties of Definite Integral

Question

$If the value of \int_{- 1}^{1} \sin^{- 1} \sqrt{| x |} \cos^{- 1} (\frac{x}{1 + x^{2 | x |}}) dx = \frac{{aπ}^{2}}{b}$

where a and b are relative prime, then (a + b) is equal to

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King and Add

$\frac{1}{2} \int_{- 1}^{1} {πsin}^{- 1} (\sqrt{| x |}) dx = \frac{π}{2} \times 2 \int_{0}^{1} \sin \sqrt{x} dx$

King and Add

$\frac{π}{2} \int_{0}^{1} (\sin^{- 1} \sqrt{x} + \sin^{- 1} \sqrt{1 - x}) dx$

$= \frac{π}{2} \int_{0}^{1} (\frac{π}{2}) dx = \frac{π^{2}}{4} \equiv \frac{{aπ}^{2}}{b}$

$∴$ a + b = 1 + 4 = 5.