Question

If f(x) = $\lim_{n \rightarrow \infty}$ [2x + 4x³ + ………. + 2nx^2n–1], (0 < x < 1) then $\int_{}^{}{f(x)dx}$is equal to –

• A. – $\sqrt{(1–x^{2})}$
• B. $\frac{1}{\sqrt{1–x^{2}}}$
• C. $\frac{1}{x^{2}–1}$
• D. $\frac{1}{1–x^{2}}$

Answer 1

Answer: (D)

$\frac{1}{1–x^{2}}$

Answer 2

f(x) = $\lim_{n \rightarrow \infty}$2 [x+ 2x³ + ……. + nx^2n–1], 0 < x < 1

= $\lim _ { n \rightarrow \infty }$ 2x [1 + 2x² + ……… + nx^2n–2]

= 2x

(Q given progression is A.G.P.)

f(x) = $\frac{2x}{(1–x^{2})^{2}}$

\ = $\int_{}^{}\frac{2x}{(1–x^{2})^{2}}$ dx

1 – x² = t

2xdx = – dt

= – $\int \frac { \mathrm { dt } } { \mathrm { t } ^ { 2 } }$

= $\frac{1}{t}$ + c

= + c