Question

Simplify:-
$\int \frac{x^2+5x+5}{x^4}dx$

Answer 1

We have,

$I=\int \frac{x^2+5x+5}{x^4}dx$

$I=\int (\frac{x^2}{x^4}+\frac{5x}{x^4}+\frac{5}{x^4})dx$

$I=\int (\frac{1}{x^2}+\frac{5}{x^3}+\frac{5}{x^4})dx$

$I=\int (x^{-2}+5x^{-3}+5x^{-4})dx$

We know that

$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$

Therefore,

$I=\frac{x^{-2+1}}{-2+1}+\frac{5x^{-3+1}}{-3+1}+\frac{5x^{-4+1}}{-4+1}+C$

$I=\frac{x^{-1}}{-1}+\frac{5x^{-2}}{-2}+\frac{5x^{-3}}{-3}+C$

$I=-(\frac{1}{x}+\frac{5}{2x^2}+\frac{5}{3x^3})+C$

Hence, this is the answer.