Integration of Power Functions
Example 1
Find $\displaystyle \int{(2x+1)^5}dx$.
\( \begin{align} \displaystyle
\int{(2x+1)^5}dx &= \dfrac{(2x+1)^{5+1}}{2(5+1)} +c \\
&= \dfrac{(2x+1)^{6}}{12} +c
\end{align} \)
Example 2
Find $\displaystyle \int{\dfrac{1}{(3x-2)^4}}dx$.
\( \begin{align} \displaystyle
\int{\dfrac{1}{(3x-2)^4}}dx &= \int{(3x-2)^{-4}}dx \\
&= \dfrac{(3x-2)^{-4+1}}{3(-4+1)} +c\\
&= \dfrac{(3x-2)^{-3}}{-9} +c\\
&= -\dfrac{1}{9(3x-2)^3} +c
\end{align} \)
Example 3
Find $\displaystyle \int{\sqrt{4x+3}}dx$.
\( \begin{align} \displaystyle
\int{\sqrt{4x+3}}dx &= \int{(4x+3)^{\frac{1}{2}}}dx \\
&= \frac{(4x+3)^{\frac{1}{2}+1}}{4\big(\frac{1}{2}+1\big)} + c \\
&= \frac{(4x+3)^{\frac{3}{2}}}{4 \times \frac{3}{2}} + c \\
&= \frac{(4x+3)^{\frac{3}{2}}}{6} + c \\
&= \frac{\sqrt{(4x+3)^3}}{6} + c
\end{align} \)
Note:
This formula works only for the power of linear functions, such as $\displaystyle \int{(2x+6)^7}dx, \int{(1-3x)^6}dx$,
but does not apply for non-linear expressions such as $\displaystyle \int{(x^2+3)^4}dx, \int{(x^3+x^2+2)^6}dx$.
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
Responses