Question

$\int {\sin }^{-1}\left(3x-4x^3\right)dx$

Answer 1

$\int$ sin^–1 (3x – 4x³)dx
Let x = sin θ
⇒ dx = cos​ θ.d​θ
& θ = sin^–1 x
$\int$ sin^–1 (3x – 4x³)dx =$\int$ sin^–1 (3 sin ​θ – 4 sin³ ​θ) . cos ​θ d​θ
= ∫ sin^–1 (sin 3​θ) . cos ​θ d​θ
$$\begin{gathered} =3\int \underset{\mathrm{I}}{\theta }.\underset{\mathrm{II}}{\cos \mathit{}\theta }d\theta \\ \mathit{=}\mathit{3}\left[\theta \int \cos \mathit{}\theta d\theta \mathit{-}\int \left\{\frac{d}{d\theta }\left(\theta \right)\mathit{-}\int \cos \mathit{}\theta \mathit{}d\theta \right\}d\theta \mathit{}\right] \\ =3\left[\theta .\sin \mathit{}\theta -\int 1.\sin \mathit{}\theta \mathit{}d\theta \right] \\ =3\left[\theta .\sin \theta +\cos \theta \right]+C \\ =3\left[\theta \mathit{.}\sin \mathit{}\theta +\sqrt{1-{\sin }^2\mathrm{\theta }}\right]+C \\ =3\left[\left({\sin }^{-1}x\right).x+\sqrt{1-x^2}\right]+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\mathit{\because }\theta \mathit{=}{\sin }^{-1}x\right) \end{gathered}$$