∫x62x+x75dx=∫x6x52+x65dx2+x6=t⇒6x5dx=dt=16∫(t−2)t5dt=16t77−t63+C=162+x677−2+x663+C

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$\int x^{6} {(2 x + x^{7})}^{5} d x =_____$

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$\frac{1}{4} [\frac{{(2 + x^{6})}^{7}}{7} - \frac{{(2 + x^{6})}^{6}}{6}] + C$

$\frac{1}{6} [\frac{{(2 + x^{6})}^{7}}{7} - \frac{{(2 + x^{6})}^{6}}{3}] + C$

$\frac{1}{6} [\frac{{(2 + x^{6})}^{7}}{7} - \frac{{(2 + x^{6})}^{6}}{6}] + C$

$\frac{1}{3} [\frac{{(2 + x^{6})}^{7}}{7} - \frac{{(2 + x^{6})}^{6}}{6}] + C$

answer is D.

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Detailed Solution

$\begin{array}{l} \int x^{6} {(2 x + x^{7})}^{5} d x \\ = \int x^{6} x^{5} {(2 + x^{6})}^{5} d x \\ 2 + x^{6} = t \Rightarrow 6 x^{5} d x = d t \\ = \frac{1}{6} \int (t - 2) t^{5} d t \\ = \frac{1}{6} [\frac{t^{7}}{7} - \frac{t^{6}}{3}] + C \\ = \frac{1}{6} [\frac{{(2 + x^{6})}^{7}}{7} - \frac{{(2 + x^{6})}^{6}}{3}] + C \end{array}$