Evaluate: (i) `int(x^3sin^(-1)x^2)/(sqrt(1-x^4))\ dx` (ii) `intx^3sin(x^4+1)\ dx`

Explanation

`(i)I=int frac{x^{3} sin ^{-1} x^{2}}{sqrt{1-x^{4}}} d x`
Let `t=sin ^{-1} x^{2} `
`frac{d t}{d x}=frac{1}{sqrt{1-x^{4}}} times 2 x `
`frac{d t}{2}=frac{x d x}{sqrt{1-x^{4}}}`
`
frac{{dt}}{{dx}}=frac{1}{sqrt{1-{x}^{4}}} times 2 {x}`
`
therefore

I=frac{1}{2} int t sin t d t `
`
I=frac{1}{2}[t(-cos t)-int 1(-cos t) d t] `
`
I=frac{1}{2}[-t(cos t)+int cos t d t] `
`
I=frac{1}{2}[-t(cos t)+sin t]+C `
`
...