Understanding Indefinite Integral of Rational Functions
Using Indefinite Integral of Rational Functions requires that the format of the expression must be 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.
$$\displaystyle \int{(ax+b)^n}dx = \dfrac{(ax+b)^{n+1}}{a(n+1)} + C \ (n \ne -1)$$
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 \( \displaystyle \int{\sqrt{3x+2}}dx \).
\( \begin{aligned} \displaystyle \require{color}
\int{\sqrt{3x+2}}dx &= \int{(3x+2)^{\frac{1}{2}}}dx &\color{red} \text{convert to index form} \\
&= \dfrac{(3x+2)^{\frac{1}{2}+1}}{3\big(\frac{1}{2}+1\big)} + C \\
&= \dfrac{(3x+2)^{\frac{3}{2}}}{\frac{9}{2}} + C \\
&= \frac{2}{9}\sqrt{(3x+2)^3} + C &\color{red} \text{don’t forget to convert back to radical form} \\
\end{aligned} \\ \)
Question 3
Find \( \displaystyle \int{\frac{1}{(5x+4)^2}}dx \).
\( \begin{aligned} \displaystyle \require{color}
\int{\frac{1}{(5x+4)^2}}dx &= \int{(5x+4)^{-2}}dx &\color{red} \text{convert to index form} \\
&= \dfrac{(5x+4)^{-2+1}}{5(-2+1)} + C \\
&= -\dfrac{(5x+4)^{-1}}{5} + C \\
&= -\frac{1}{5(5x+4)} + C &\color{red} \text{convert to positive index} \\
\end{aligned} \\ \)
Question 4
Find \( \displaystyle \int{\frac{1}{\sqrt{6x-1}}}dx \).
\( \begin{aligned} \displaystyle \require{color}
\int{\frac{1}{\sqrt{6x-1}}}dx &= \int{(6x-1)^{-\frac{1}{2}}}dx &\color{red} \text{convert to index form} \\
&= \dfrac{(6x-1)^{-\frac{1}{2}+1}}{6(-\frac{1}{2}+1)} + C \\
&= \dfrac{(6x-1)^{\frac{1}{2}}}{3} + C \\
&= \frac{1}{3}\sqrt{6x-1} + C &\color{red} \text{convert back to radical form} \\
\end{aligned} \\ \)
Question 5
Find \( \displaystyle \int{\sqrt[3]{(2-3x)^4}}dx \).
\( \begin{aligned} \displaystyle \require{color}
\int{\sqrt[3]{(2-3x)^4}}dx &= \int{(2-3x)^{\frac{4}{3}}}dx &\color{red} \text{convert to index form} \\
&= \dfrac{(2-3x)^{\frac{4}{3}+1}}{-3\big(\frac{4}{3}+1\big)} + C \\
&= \dfrac{(2-3x)^{\frac{7}{3}}}{-7} + C \\
&= -\frac{1}{7} \sqrt[3]{(2-3x)^{7}} + C &\color{red} \text{convert back to radical form} \\
\end{aligned} \\ \)
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Chain Rule Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials 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 Proof Pythagoras Theorem Quadratic Quadratic Factorise Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume