$\int (1 + 2 x + 3 x^{2} + 4 x^{3} + . . . . .) d x (| x | < 1)$

- (1 + x)^{- 1} + c

No worries! We‘ve got your back. Try BYJU‘S free classes today!

(1 - x)^{- 1} + c

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

- (1 - x)^{- 2} + c

No worries! We‘ve got your back. Try BYJU‘S free classes today!

none of these

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $(1 - x)^{- 1} + c$
Let $S = 1 + 2 x + {3 x}^{2} + 4 x^{3} + . . .$ ...(1)
Multiply this by $x$
$x S = x + {2 x}^{2} + {3 x}^{3} + . . .$ ...(2)
(1) -(2) gives
$(1 - x) S = 1 + x + x^{2} + x^{3} + . . .$
$\Rightarrow (1 - x) S = \frac{1}{1 - x} \Rightarrow S = \frac{1}{{(1 - x)}^{2}}$
$∴ \int S d x = \int \frac{1}{{(1 - x)}^{2}} d x = {(1 - x)}^{- 1} + c$

Suggest Corrections

Similar questions

1 + 2 x + 3 x^{2} + 4 x^{3} + . . . . . . \infty

\int (1 + 2 x + 3 x^{2} + 4 x^{3} + . . .) d x (0 < | x | < 1)

\int (1 + 2 x + 3 x^{2} + 4 x^{3} + \dots) d x, (0 < | x | < 1)

\int (1 + 2 x + 3 x^{2} + 4 x^{3} + . . . . . .) d x

(| x | < 1)

1 + 2 x + 3 x^{2} + 4 x^{3} + . . . . . \infty; | x | < 1

Join BYJU'S Learning Program