Question

Evaluate: $\int \frac{x^3+5x^2+4x+1}{x^2}dx$.

Answer 1

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

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

Now integral of $\int x^{a}dx=\frac{x^{a+1}}{a+1}$ and of $\int \frac{1}{x}dx=\ln x$

Applying this in $I$ we get,

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