Evaluate: (i) `int((x+1)e^x)/(cos^2(x e^x))\ dx` (ii) `intx^2\ e^x^3cos(e^x^3)\ dx`

Explanation

(i)
`int frac{(x+1) e^{x}}{cos ^{2}(x cdot e^{x})} d x `

`text { Let } {x} {e}^{x}=t `

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

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

Now, `int frac{(x+1) e^{x}}{cos ^{2}(x cdot e^{x})} d x `

`=int frac{d t}{cos ^{2} t} `

`=int sec ^{2} {t} d t `

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