# Indefinite Integral of Rational Functions

## Understanding the Indefinite Integral of Rational Functions

Using Indefinite Integral of Rational Functions requires that the format of the expression must be the power of linear expressions, such as $(3x-1)^3, (2x+3)^3, \sqrt{4x-1}$, etc. $(3x^2-1)^3, (2\sqrt{x}+3)^3, \sqrt{4x^3-1}$ are not applicable for this formula.
$$\large \displaystyle \int{(ax+b)^n}dx = \dfrac{(ax+b)^{n+1}}{a(n+1)} + C \ (n \ne-1)$$

# Practice Questions

### Question 1

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

\begin{aligned} \displaystyle \int{(2x-1)^3}dx &= \dfrac{(2x-1)^{3+1}}{2(3+1)} + C \\ &= \frac{1}{8}(2x-1)^4 + C \end{aligned}

### Question 2

Find $\displaystyle \int{\sqrt{3x+2}}dx$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \int{\sqrt{3x+2}}dx &= \int{(3x+2)^{\frac{1}{2}}}dx &\color{red} \text{convert to index form} \\ &= \dfrac{(3x+2)^{\frac{1}{2}+1}}{3\big(\frac{1}{2}+1\big)} + C \\ &= \dfrac{(3x+2)^{\frac{3}{2}}}{\frac{9}{2}} + C \\ &= \frac{2}{9}\sqrt{(3x+2)^3} + C &\color{red} \text{don’t forget to convert back to radical form} \end{aligned}

### Question 3

Find $\displaystyle \int{\frac{1}{(5x+4)^2}}dx$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \int{\frac{1}{(5x+4)^2}}dx &= \int{(5x+4)^{-2}}dx &\color{red} \text{convert to index form} \\ &= \dfrac{(5x+4)^{-2+1}}{5(-2+1)} + C \\ &= -\dfrac{(5x+4)^{-1}}{5} + C \\ &= -\frac{1}{5(5x+4)} + C &\color{red} \text{convert to positive index} \end{aligned}

### Question 4

Find $\displaystyle \int{\frac{1}{\sqrt{6x-1}}}dx$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \int{\frac{1}{\sqrt{6x-1}}}dx &= \int{(6x-1)^{-\frac{1}{2}}}dx &\color{red} \text{convert to index form} \\ &= \dfrac{(6x-1)^{-\frac{1}{2}+1}}{6(-\frac{1}{2}+1)} + C \\ &= \dfrac{(6x-1)^{\frac{1}{2}}}{3} + C \\ &= \frac{1}{3}\sqrt{6x-1} + C &\color{red} \text{convert back to radical form} \end{aligned}

### Question 5

Find $\displaystyle \int{\sqrt[3]{(2-3x)^4}}dx$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \int{\sqrt[3]{(2-3x)^4}}dx &= \int{(2-3x)^{\frac{4}{3}}}dx &\color{red} \text{convert to index form} \\ &= \dfrac{(2-3x)^{\frac{4}{3}+1}}{-3\big(\frac{4}{3}+1\big)} + C \\ &= \dfrac{(2-3x)^{\frac{7}{3}}}{-7} + C \\ &= -\frac{1}{7} \sqrt[3]{(2-3x)^{7}} + C &\color{red} \text{convert back to radical form} \\ \end{aligned}

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