Integration of Power Functions

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 $\displaystyle \int{\sqrt{4x+3}}dx$.

\( \begin{align} \displaystyle
\int{\sqrt{4x+3}}dx &= \int{(4x+3)^{\frac{1}{2}}}dx \\
&= \frac{(4x+3)^{\frac{1}{2}+1}}{4\big(\frac{1}{2}+1\big)} + c \\
&= \frac{(4x+3)^{\frac{3}{2}}}{4 \times \frac{3}{2}} + c \\
&= \frac{(4x+3)^{\frac{3}{2}}}{6} + c \\
&= \frac{\sqrt{(4x+3)^3}}{6} + c
\end{align} \)

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 $\displaystyle \int{(x^2+3)^4}dx, \int{(x^3+x^2+2)^6}dx$.


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