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Evaluate: intcos^(-1)((1-x^2)/(1+x^2))\ ...

Evaluate: `intcos^(-1)((1-x^2)/(1+x^2))\ dx`

Text Solution

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`Let {I}=int cos ^{-1}(frac{1-x^{2}}{1+x^{2}}) {dx}`
` =2 int 1_{I I} cdot tan _{I}^{-1} {x} {dx}[because cos ^{-1}(frac{1-x^{2}}{1+x^{2}})=2 tan ^{-1} x] `
` =2[tan ^{-1} x int 1 {dx}-int{frac{d}{d x}(tan ^{-1} x) int 1 {dx}} d x] `
` =2[tan ^{-1} x cdot x-int frac{1}{1+x^{2}} times x {dx}]`
` =2 x tan ^{-1} x-int frac{2 x}{1+x^{2}} {dx}`
Putting` 1+x^{2}=t`
` Rightarrow 2 {dx}=d t`
` therefore I=2 x tan ^{-1} x-int frac{d t}{t}`
` ...
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