First slide

Introduction to integration

Question

$\int \frac{p x^{p + 2 q - 1} - q x^{q - 1}}{x^{2 p + 2 q} + 2 x^{p + q} + 1} d x =$

Moderate

A

$- \frac{x^{p}}{x^{p + q} + 1} + c$
B

$\frac{x^{q}}{x^{p + q} + 1} + c$
C

$- \frac{x^{q}}{x^{p + q} + 1} + c$
D

$\frac{x^{p}}{x^{p + q} + 1} + c$

Solution

$\begin{array}{l} \int \frac{p x^{p + 2 q - 1} - q x^{q - 1}}{{(x^{p + q} + 1)}^{2}} d x \\ D i v i d i n g n u m e r a t o r a n d d e n o m i n a t o r b y x^{2 q} \\ = \int \frac{p x^{p - 1} - q x^{- q - 1}}{{(x^{p} + x^{- q})}^{2}} d x \\ (\begin{array}{l} p u t x^{p} + x^{- q} = t \\ \Rightarrow (p x^{p - 1} - q x^{- q - 1}) d x = d t \end{array}) \\ = \int \frac{d t}{t^{2}} = \frac{- 1}{t} + c \\ = \frac{- 1}{x^{p} + x^{- q}} + c \\ = \frac{- x^{q}}{x^{p + q} + 1} + c \end{array}$

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