# Integration of Power Functions

### Example 1

Find $\displaystyle \int{(2x+1)^5}dx$.

\begin{align} \displaystyle \int{(2x+1)^5}dx &= \dfrac{(2x+1)^{5+1}}{2(5+1)} +c \\ &= \dfrac{(2x+1)^{6}}{12} +c \end{align}

### Example 2

Find $\displaystyle \int{\dfrac{1}{(3x-2)^4}}dx$.

\begin{align} \displaystyle \int{\dfrac{1}{(3x-2)^4}}dx &= \int{(3x-2)^{-4}}dx \\ &= \dfrac{(3x-2)^{-4+1}}{3(-4+1)} +c\\ &= \dfrac{(3x-2)^{-3}}{-9} +c\\ &= -\dfrac{1}{9(3x-2)^3} +c \end{align}

### Example 3

Find $\displaystyle \int{\sqrt{4x+3}}dx$.

\begin{align} \displaystyle \int{\sqrt{4x+3}}dx &= \int{(4x+3)^{\frac{1}{2}}}dx \\ &= \frac{(4x+3)^{\frac{1}{2}+1}}{4\big(\frac{1}{2}+1\big)} + c \\ &= \frac{(4x+3)^{\frac{3}{2}}}{4 \times \frac{3}{2}} + c \\ &= \frac{(4x+3)^{\frac{3}{2}}}{6} + c \\ &= \frac{\sqrt{(4x+3)^3}}{6} + c \end{align}

Note:
This formula works only for the power of linear functions, such as $\displaystyle \int{(2x+6)^7}dx, \int{(1-3x)^6}dx$,
but does not apply for non-linear expressions such as $\displaystyle \int{(x^2+3)^4}dx, \int{(x^3+x^2+2)^6}dx$.