Question

If $f\left(x\right)=\underset{n\to \infty }{lim}[2x+4x^3+\dots \dots .+2n{x}^{2n-1}](0<x<1)$ then $\int f\left(x\right)dx$ is equal to

• A. $-\sqrt{1-x^2}+constant$
• B. $\frac{1}{\sqrt{1-x^2}}+constant$
• C. $\frac{1}{x^2-1}+constant$
• D. $\frac{1}{1-x^2}+constant$

Answer 1

Answer: (D)

$\frac{1}{1-x^2}+constant$

$\int f\left(x\right)dx=\underset{n\to \infty }{lim}[\int (2x+4x^3+.............+2n{x}^{2n-1}\left)dx\right]$

$=\underset{n\to \infty }{lim}(x^2+x^4+x^6+.................x^{2n}+c)=\frac{x^2}{1-x^2}+c$

$=-\frac{1-x^2-1}{1-x^2}+c=\frac{1}{1-x^2}+(c-1)=\frac{1}{1-x^2}+K$ With $K=constant$