Question

Solve $\int \left(1+2x+3x^2+4x^3+...\right)dx\left(0<\left|x\right|<1\right)$

• A. $\frac{x\left(x^{n-1}\right)}{x-1}+C$
• B. $\frac{x(x^n-1)}{x-1}+C$
• C. $\frac{(x^n-1)}{x-1}+C$
• D. $\frac{x(x^n+1)}{x+1}+C$

Answer 1

Answer: (B)

$\frac{x(x^n-1)}{x-1}+C$

$\int (1+2x+{3x}^2+{4x}^3+.........{nx}^n)dx$

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

$=x+\frac{2x^2}{2}+\frac{3x^3}{3}+............\frac{(n+1)x^{n+1}}{n+1}$

$=x+x^2+x^3+..........x^{n+1}+C$.......(1)

It's a G.P. having first term=$\left(x\right)$

and

common ratio=$\left(x\right)$

then eqn. (1) becomes

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

(by using formula for sum of G.P when $S_n=\left(\frac{r^n-1}{r-1}\right)$)

Hence, the answer is $\frac{x(x^n-1)}{(x-1)}+C$

Hence, the option is $B$