If _ ^ ( x^2 + x)dx = x ( x^2 + x) + A , then A = - Vidyadip

MCQ

If $\int_{}^{} {\ln ({x^2} + x)dx = x\ln ({x^2} + x) + A} $, then $A = $

A
$2x + \ln (x + 1) + $constant
B
$2x - \ln (x + 1) + $constant
C
Constant
✓
None of these

✓

Answer

Correct option: D.

None of these

d
(d)$\int_{}^{} {\log ({x^2} + x)\,dx} = \int_{}^{} {\log x\,dx} + \int_{}^{} {\log (x + 1)\,dx} $
$ = x\log x - x + x\log (x + 1) - x + \log (x + 1)$
$ = x\left\{ {(\log x + \log (x + 1)} \right\} - 2x + \log (x + 1)$
$ = x\log ({x^2} + x) - 2x + \log (x + 1)$
Equating it to the given integration, we get
$A = - 2x + \log (x + 1)$.

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