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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