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Evaluate : int0^1 1/(1+2x+2x^2+2x^3+x^4...

Evaluate : `int_0^1 1/(1+2x+2x^2+2x^3+x^4)dx`

Text Solution

Verified by Experts

`I=int_{0}^{1} frac{1}{1+2 {x}+2 {x}^{2}+2 {x}^{3}+{x}^{4}} d {x}`
BY arranging denominator, we recieve
` 1+2 x+2 x^{2}+2 x^{3}+x^{4}=(1+x)^{2}(x^{2}+1) `
` int_{0}^{1} frac{1}{(1+x)^{2}(x^{2}+1)} d x `
` =int_{0}^{1} frac{-x}{2(x^{2}+1)} d x+int_{0}^{1} frac{1}{2(x+1)} d x `
` +int_{0}^{1} frac{1}{2(1+x)^{2}} d x`
...
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