Evaluate: (i) `int1/(x^2(x^4+1)^(3//4))\ dx` (ii) `int(sin^5x)/(cos^4x)\ dx`

Explanation

(i)
`int frac{d x}{x^{2}(x^{4}+1)^{frac{3}{4}}} `

`=int frac{d x}{x^{2}[x^{4}(1+frac{1}{x^{4}})]^{frac{3}{4}}} `

`=int frac{d x}{x^{2} cdot x^{3}(1+frac{1}{x^{4}})^{frac{3}{4}}} `

`=int frac{(1+frac{1}{x^{4}})^{-frac{3^{4}}{4}}}{x^{5}} dx `

`text { Let } 1+frac{1}{x^{4}}=t `

`Rightarrow-frac{4}{x^{5}} d x=dt `

`Rightarrow frac{d x}{x^{5}}=-frac{d t}{4} `

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