Using integration by parts, find $\displaystyle\int{{{\sin }^{-1}}}2x\ \mathrm{d}x$. Hence evaluate

Question

Using integration by parts, find $\displaystyle\int{{{\sin }^{-1}}}2x\ \mathrm{d}x$. Hence evaluate $\displaystyle\int_{-\frac{1}{2}}^{\frac{1}{4}}{\left| {{\sin }^{-1}}2x \right|}\ \mathrm{d}x$ exactly.

Accepted Answer

(i) $x\sin^{-1}(2x) + \frac{1}{2}\sqrt{1-4x^2} + C$ (ii) $\frac{7}{24}\pi + \frac{\sqrt{3}}{4} - 1$

Integration of Inverse Trigonometric Function Involving Modulus

Integration of Inverse Trigonometric Function Involving Modulus

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