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Lesson 7, Topic 5
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# Function of Linear Function of x

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When the variable x replaced by a linear expression in x, i.e. of the form ax + b such that we have (ax + b)n instead of xn , we evaluate the integral by  substitution.

Example:

Integrate (3x + 2)4

$$\scriptsize \int (3x + 2)^4dx, \;\\ \scriptsize let \; u = 3x +2$$

$$\scriptsize \int u^4 dx$$ we find x and substitute, $$\frac{du}{dx} \scriptsize = 3, \therefore dx =\normalsize \frac{du}{3}$$

$$\scriptsize \int u^4 \frac{du}{3} = \frac {1}{3}\int u^4 du = \frac {1}{3}\left( \frac{u^5}{5} \right) + c$$

Substituting u = 3x +2

= $$\frac {1}{3}\left( \frac{(3x + 2)^5}{5} \right) + c$$

= $$\frac{(3x + 2)^5}{15} + c$$

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