Radical Indefinite Integrals should be performed after converting its radical or surd notations into index form.
$$\large \displaystyle \sqrt[n]{x^m} = x^{\frac{m}{n}}$$

# Practice Questions

### Question 1

Find $\displaystyle \int{\sqrt{x}}dx$.

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

### Question 2

Find $\displaystyle \int{\sqrt[3]{x^5}}dx$.

\begin{aligned} \displaystyle \int{\sqrt[3]{x^5}}dx &= \int{x^{\frac{5}{3}}}dx \\ &= \dfrac{x^{\frac{5}{3}+1}}{\frac{5}{3}+1} + C \\ &= \dfrac{x^{\frac{8}{3}}}{\frac{8}{3}} + C \\ &= \frac{3}{8}\sqrt[3]{x^8} + C \end{aligned}

### Question 3

Find $\displaystyle \int{\sqrt{5x^3}}dx$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \int{\sqrt{5x^3}}dx &= \sqrt{5} \int{\sqrt{x^3}}dx &\color{red} \text{separate the coefficient} \\ &= \sqrt{5}\int{x^{\frac{3}{2}}}dx \\ &= \sqrt{5} \times \dfrac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1} + C \\ &= \sqrt{5} \times \dfrac{x^{\frac{5}{2}}}{\frac{5}{2}} + C \\ &= \frac{2\sqrt{5}}{5} \sqrt{x^5}+ C \end{aligned}

### Question 4

Find $\displaystyle \int{\frac{3x-1}{\sqrt{x}}}dx$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \int{\frac{3x-1}{\sqrt{x}}}dx &= \int{\Bigg(\frac{3x}{\sqrt{x}}-\frac{1}{\sqrt{x}}\Bigg)}dx &\color{red} \text{separate into two single fractions} \\ &= \int{\Big(3x^{\frac{1}{2}}-x^{-\frac{1}{2}}\Big)}dx + C \\ &= \dfrac{3x^{\frac{1}{2}+1}}{\frac{1}{2}+1}-\dfrac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} + C \\ &= \dfrac{3x^{\frac{3}{2}}}{\frac{3}{2}}-\dfrac{x^{\frac{1}{2}}}{\frac{1}{2}} + C \\ &= 2\sqrt{x^3}-2\sqrt{x} + C \end{aligned}

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