Radical Indefinite Integrals

Radical Indefinite Integrals

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 \( \displaystyle \int{\frac{3x-1}{\sqrt{x}}}dx \).

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

Unlock your full learning potential—download our expertly crafted slide files for free and transform your self-study sessions!

Discover more enlightening videos by visiting our YouTube channel!

 

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume




Related Articles

Responses

Your email address will not be published. Required fields are marked *