Question

Evaluate : $\int (1+2x+3x^2+4x^3+\dots )dx,(0<|x|<1)$

Answer 1

$s=1+2x+3x^2+4x^3+5x^4+...$

$xs=x+2x^2+3x^3+4x^4+5x^5$

$s(1-x)=1+x+x^2+x^3+x^4+x^5+$

$s(1-x)=\frac{1}{1-x}$ sum of infinite terms in G.P.

$s=\frac{1}{(1-x)2}$

Form equation-

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

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

$=\int (1-x{)}^{-2}dx.$

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

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