Evaluate: (i) `int(e^(sqrt(x))cos(e^(sqrt(x))))/(sqrt(x))\ dx` (ii) `int(cos^5x)/(sinx)\ dx`

Explanation

(i)
`int frac{e^{sqrt{x}} cdot cos (e^{sqrt{x}})}{sqrt{x}} d x `

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

`Rightarrow e^{sqrt{x}} times frac{1}{2 sqrt{x}}=frac{d t}{d x} `

`Rightarrow frac{e^{sqrt{x}}}{sqrt{x}} d x=2 d t `

Now, `int frac{e^{sqrt{x}} cdot cos (e^{sqrt{x}})}{sqrt{x}} d x `

`=2 int cos t {dt} `

`=2 sin t+C `

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