Evaluate: (i) `int1/(a^2\ x^2-b^2)\ dx` (ii) `int(x^2-1)/(x^2+4)\ dx`

Explanation

(i)
`int frac{d x}{a^{2} x^{2}+b^{2}} `

`= frac{1}{a^{2}} int frac{d x}{x^{2}+(frac{b}{a})^{2}} `

`= frac{1}{a^{2}} times frac{a}{b} tan ^{-1}(frac{x}{frac{b}{a}})+C[therefore int frac{d x}{a^{2}+x^{2}}=frac{1}{a} tan ^{-1}(frac{x}{a})+C] `

`= frac{1}{a b} tan ^{-1}(frac{a x}{b})+C`


(ii)
`int frac{x^{2}-1}{x^{2}+4} d x `

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

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