First slide

Methods of integration

Question

$\int (x + 5) (x + 1) {(x + 3)}^{5} d x =_____$

Moderate

A

$\frac{7}{8} {(x + 2)}^{8} - \frac{2}{3} {(x + 3)}^{6} + C$
B

$\frac{7}{8} {(x + 2)}^{7} + \frac{2}{3} {(x + 3)}^{6} + C$
C

$\frac{{(x + 2)}^{8}}{8} + \frac{2}{3} {(x + 3)}^{6} + C$
D

$\frac{{(x + 2)}^{8}}{8} - \frac{2}{3} {(x + 3)}^{6} + C$

Solution

$\begin{array}{l} I = \int (x + 3 + 2) (x + 3 - 2) (x + 3)^{5} d x \\ Let x + 3 = t \Rightarrow d x = d t \\ I = \int (t + 2) (t - 2) t^{5} d t \\ I = \int t^{7} - 4 t^{5} d t \\ \frac{t^{8}}{8} - \frac{4 t^{6}}{6} + C \end{array}$

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