Question

Evaluate : $\int \frac{dx}{x^4+x^3}$

Answer 1

$I=\int \frac{dx}{x^4+x^3}$

$\Rightarrow I=\int \frac{dx}{x^3(x+1)}$

$\Rightarrow \frac{1}{x^3(x+1)}$

$\Rightarrow \frac{-1}{x+1}+\frac{x^2+1}{x^3}$

$\Rightarrow 1=\int (\frac{-1}{x+1}+\frac{x^2-x+1}{x^2})dx$

$=\int (\frac{-1}{x+1}+\frac{1}{x}-\frac{1}{x^2}+\frac{1}{x^3})dx$

$=-\ln (x+1)+\ln x+\frac{1}{x}-\frac{1}{2x^2}+c$

$\Rightarrow I=\ln \left(\frac{x}{x+1}\right)+\frac{1}{x}-\frac{1}{2x^2}+c$

Hence, the answer is $\ln \left(\frac{x}{x+1}\right)+\frac{1}{x}-\frac{1}{2x^2}+c.$