# Definite Integration of Power Functions

$$\large \displaystyle \int_{n}^{m}{(ax+b)^k}dx = \dfrac{1}{a(k+1)}\Big[(ax+b)^{k+1}\Big]_{n}^{m}+c$$

### Example 1

Find $\displaystyle \int_{0}^{1}{(2x+1)^5}dx$.

\begin{align} \displaystyle \int{(2x+1)^5}dx &= \left[\dfrac{(2x+1)^{5+1}}{2(5+1)}\right]_{0}^{1} \\ &= \dfrac{1}{12}\big[(2x+1)^{6}\big]_{0}^{1} \\ &= \dfrac{1}{12}\big[(2 \times 1+1)^{6}-(2 \times 0+1)^{6}\big] \\ &= \dfrac{1}{12}(729-1) \\ &= \dfrac{728}{12} \\ &= \dfrac{182}{3} \end{align}

### Example 2

Find $\displaystyle \int_{0}^{1}{\dfrac{1}{(3x-2)^4}}dx$.

\begin{align} \displaystyle \int_{0}^{1}{\dfrac{1}{(3x-2)^4}}dx &= \int_{0}^{1}{(3x-2)^{-4}}dx \\ &= \bigg[\dfrac{(3x-2)^{-4+1}}{3(-4+1)}\bigg]_{0}^{1} \\ &= \bigg[\dfrac{(3x-2)^{-3}}{-9}\bigg]_{0}^{1} \\ &= -\dfrac{1}{9}\big[(3x-2)^3\big]_{0}^{1} \\ &= -\dfrac{1}{9}\big[(3 \times 1-2)^3-(3 \times 0-2)^3\big] \\ &= -\dfrac{1}{9}(1 +8) \\ &= -1 \end{align}

### Example 3

Find $\displaystyle \int_{0}^{1}{\sqrt{4x+3}}dx$.

\begin{align} \displaystyle \int{\sqrt{4x+3}}dx &= \int_{0}^{1}{(4x+3)^{\frac{1}{2}}}dx \\ &= \bigg[\frac{(4x+3)^{\frac{1}{2}+1}}{4\big(\frac{1}{2}+1\big)}\bigg]_{0}^{1} \\ &= \bigg[\frac{(4x+3)^{\frac{3}{2}}}{4 \times \frac{3}{2}}\bigg]_{0}^{1} \\ &= \bigg[\frac{(4x+3)^{\frac{3}{2}}}{6}\bigg]_{0}^{1} \\ &= \frac{1}{6}\Big[\sqrt{(4x+3)^3}\Big]_{0}^{1} \\ &= \frac{1}{6}\Big[\sqrt{(4 \times 1+3)^3}-\sqrt{(4 \times 0+3)^3}\Big] \\ &= \frac{\sqrt{7^3}-\sqrt{3^3}}{6} \end{align}

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