Question

Evaluate: $\int (1+2x+3x^2+4x^3+....)$ dx

Answer 1

$\int (1+2x+3x^2+4x^3+....)dx$

$=\int dx+\int 2xdx+\int 3x^{2}dx+\int 4x^{3}dx+....$

$=x+\frac{2x^2}{2}+\frac{3x^3}{3}+\frac{4x^4}{4}+...$

$=x+x^2+x^3+x^4+...$

$=x(1+x+x^2+x^3+...)$

$1+x+x^2+x^3+...$ are in G.P whose first term is $1$ and common ratio is $\frac{x}{1}=x$

$\therefore S_{\infty }=\frac{a}{1-r}=\frac{1}{1-x}$

$\therefore \int (1+2x+3x^2+4x^3+....)dx=x\left(\frac{1}{1-x}\right)=\frac{x}{1-x}$