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

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Solution

Let,

$S = 1 + 2 x + 3 x^{2} + 4 x^{3} + . . . \dots$

$S x = x + 2 x^{2} + 3 x^{3} + . . . . . .$

So, $S - S x = 1 + x + x^{2} + x^{3} + . . . \dots$

$⟹ S (1 - x) = \frac{1}{1 - x}$

$⟹ S = \frac{1}{(1 - x)^{2}}$

$⟹ \int S = \int \frac{1}{(1 - x)^{2}} d x$

$⟹ \frac{1}{1 - x} + C$

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