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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The text passage contains multiple mathematical questions, each with a corresponding answer provided. The questions are about finding indefinite integrals involving radical expressions, and the answers show the step-by-step solutions.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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}

## Mastering Integration by Parts: The Ultimate Guide

Welcome to the ultimate guide on mastering integration by parts. If you’re a student of calculus, you’ve likely encountered integration problems that seem insurmountable. That’s…

## Trigonometry Made Easy: Integration by Parts Demystified

Integration by Parts is made of the product rule of differentiation. The derivative of $uv$ is $u’v + uv’$ and integrates both sides. \( \begin{aligned}…

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## 12 Patterns of Logarithmic Equations

Solving logarithmic equations is done using properties of logarithmic functions, such as multiplying logs and changing the base and reciprocals of logarithms.  \large \begin{aligned}…
user
The text passage contains multiple mathematical questions, each with a corresponding answer provided. The questions are about finding indefinite integrals involving radical expressions, and the answers show the step-by-step solutions.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS