First slide

Introduction to integration

Question

$\int \frac{7 x^{13} + 5 x^{15}}{{(x^{7} + x^{2} + 1)}^{3}}$ equals

Easy

A

$\frac{x^{14}}{{(x^{7} + x^{2} + 1)}^{2}} + C$
B

$\frac{x^{14}}{2 {(x^{7} + x^{2} + 1)}^{2}} + C$
C

$\frac{x^{7}}{{(x^{7} + x^{2} + 1)}^{2}} + C$
D

$\frac{x^{7}}{2 {(x^{7} + x^{2} + 1)}^{2}} + C$

Solution

$\begin{array}{l} \int \frac{7 x^{13} + 5 x^{15}}{{(x^{7} + x^{2} + 1)}^{3}} d x \\ = \int \frac{7 x^{13} (x^{7} + x^{2} + 1) - x^{14} (7 x^{6} + 2 x)}{{(x^{7} + x^{2} + 1)}^{3}} d x \\ = \int \frac{d}{d x} (\frac{x^{14}}{2 {(x^{7} + x^{2} + 1)}^{2}}) d x \\ = \frac{x^{14}}{2 {(x^{7} + x^{2} + 1)}^{2}} + C \end{array}$

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