Function of Linear Function of x

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

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

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