First slide

Methods of integration

Question

$\int \frac{7 x^{8} + 8 x^{7}}{{(1 + x + x^{8})}^{2}} d x =$

Moderate

A

$\frac{4 x^{8}}{1 + x + x^{8}} + c$
B

$\frac{x^{8}}{{(1 + x + x^{8})}^{2}} + c$
C

$\frac{2 x^{8}}{{(1 + x + x^{8})}^{2}} + c$
D

$\frac{x^{8}}{1 + x + x^{8}} + c$

Solution

Consider
$\begin{array}{l} \frac{d}{d x} (\frac{x^{8}}{1 + x + x^{8}}) = \frac{(1 + x + x^{8}) 8 x^{7} - x^{8} (1 + 8 x^{7})}{{(1 + x + x^{8})}^{2}} \\ = \frac{8 x^{7} + 8 x^{8} + 8 x^{5} - x^{8} - 8 x^{15}}{{(1 + x + x^{8})}^{2}} \\ = \frac{7 x^{8} + 8 x^{7}}{{(1 + x + x^{8})}^{2}} \end{array}$
$\begin{array}{l} Now \int \frac{7 x^{8} + 8 x^{7}}{{(1 + x + x^{8})}^{2}} d x = \int \frac{d}{d x} (\frac{x^{8}}{(1 + x + x^{8})}) \\ = \frac{x^{8}}{(1 + x + x^{8})} + c \end{array}$

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