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Evaluate : int("dx")/(1+"x"+"x"^2+"x"^...

Evaluate : `int("dx")/(1+"x"+"x"^2+"x"^3)`

Text Solution

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`I=int frac{d x}{1+x+x^{2}+x^{3}} `
`=int frac{d x}{(1+x)+x^{2}(1+x)} `
`=int frac{d x}{(x+1)(x^{2}+1)} `
`text { Let } frac{1}{(x+1)(x^{2}+1)}=frac{A}{x+1}+frac{B x+C}{x^{2}+1} `
`Rightarrow frac{1}{(x+1)(x^{2}+1)}=frac{A(x^{2}+1)+(B x+C)(x+1)}{(x+1)(x^{2}+1)} `
`Rightarrow 1=A(x^{2}+1)+B x^{2}+B x+C x+C `
`Rightarrow 1=(A+B) x^{2}+(B+C) x+(A+C)`
Equating coefficients of like terms
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