If x 3 x + 1 ? d x = q ( x ) - log | x + 1 | + C , then q ( x ) is equal to:

Question

If $\int \frac{x^{3}}{x + 1} d x = q (x) - \log$ $| x + 1 |$ $+ C$ , then $q (x)$ is equal to:

Smart Achievers · Accepted Answer

(4)

Ans. $q (x) = \frac{x^{3}}{3} - \frac{x^{2}}{2} + x$

Explanation :

$\int \frac{x^{3}}{x + 1} d x = \int \frac{x^{3} + 1 - 1}{x + 1} d x$

$= \int \frac{x^{3} + 1}{x + 1} d x - \int \frac{1}{x + 1} d x$

$= \int (x^{2} + 1 - x) d x - \log (x + 1)$

$= \frac{x^{3}}{3} + x - \frac{x^{2}}{2} - \log$ $| x + 1 |$ $+ C$

$So$ $q (x) = \frac{x^{3}}{3} - \frac{x^{2}}{2} + x$

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