Question
Integrate the following functions w.r.t. x:
$x^2\left(1-\frac{2}{x}\right)^2$
$x^2\left(1-\frac{2}{x}\right)^2$
✓
Answer
$\quad \int x^2\left(1-\frac{2}{x}\right)^2 d x=\int x^2\left(1-\frac{4}{x}+\frac{4}{x^2}\right) d x$
$=\int\left(x^2-4 x+4\right) d x$
$=\int x^2 d x-4 \int x d x+4 \int 1 d x$
$=\frac{x^3}{3}-4\left(\frac{x^2}{2}\right)+4 x+c$
$=\frac{1}{3} x^3-2 x^2+4 x+c$
$=\int\left(x^2-4 x+4\right) d x$
$=\int x^2 d x-4 \int x d x+4 \int 1 d x$
$=\frac{x^3}{3}-4\left(\frac{x^2}{2}\right)+4 x+c$
$=\frac{1}{3} x^3-2 x^2+4 x+c$
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