Evaluate: (i) `int5^x+tan^((-1)x)((x^2+2)/(x^2+1))\ dx` (ii) `int(e^msin^((-1)x))/(sqrt(1-x^2))\ dx`

Explanation

(i)
`int 5^{x+tan ^{-1} x} cdot(frac{x^{2}+2}{x^{2}+1}) d x `

`text { Let } x+tan ^{-1} x=t `

`(1+frac{1}{1+x^{2}})=frac{d t}{d x} `

`Rightarrow(frac{x^{2}+1+1}{x^{2}+1}) d x=d t `

`Rightarrow(frac{x^{2}+2}{x^{2}+1}) d x=d t `

Now, `int 5^{x+tan ^{-1} x} cdot(frac{x^{2}+2}{x^{2}+1}) d x `

`=int 5^{t} d t `

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