The antiderivative of a power function

Integration: Antiderivatives

The antiderivative of a power function

As with differentiation, we use rules to determine the antiderivative of standard functions when integrating. We will first look at the antiderivative of a power function.

For #\orange n\neq -1#:

\[\int x^\orange {n}\;\dd x = \frac{1}{\orange n+1}x^{\orange n+1} + \green C \]

Example

#\begin{array}{rcl}
\displaystyle \int x^\orange 4 \;\dd x &=&\dfrac{1}{\orange 4+1}x^{\orange 4+1} + \green C \\
&=&\dfrac 15 x^5 + \green C
\end{array}#

We can prove this by showing that if we differentiate the antiderivative #\frac{1}{\orange n+1}x^{\orange n+1} + \green C#, we indeed get the original function #x^\orange {n}# again.

\[\begin{array}{rcl}\frac{\dd}{\dd x}\left(\frac{1}{\orange n+1}x^{\orange n+1} + \green C\right)&=& \frac{\dd}{\dd x}\left(\frac{1}{\orange n+1}x^{\orange n+1}\right)+\frac{\dd}{\dd x} \green C \\ &&\phantom{xxx}\blue{\text{sum rule}} \\ &=&\frac{1}{\orange n+1} \frac{\dd}{\dd x}x^{\orange n+1} +0 \\ &&\phantom{xxx}\blue{\text{constant rule and derivative constant is }0} \\ &=& \frac{1}{\orange n+1} \cdot \left(\orange n+1\right) \cdot x^{\orange n +1-1} \\ &&\phantom{xxx}\blue{\text{power rule }\frac{\dd}{\dd x}x^n=n \cdot x^{n-1}} \\ &=& x^{\orange n}\\ &&\phantom{xxx}\blue{\text{simplified}}\end{array}\]

Root

We can also determine the antiderivative of a root using this rule.

\[\int \sqrt{x} \;\dd x = \int x^\orange{\frac{1}{2}} \; \dd x = \frac{1}{\orange{\frac{1}{2}}+1}x^{\orange{\frac{1}{2}}+1} +\green C= \frac{2}{3}x \sqrt{x}+\green C\]

Determine an antiderivative #F# for the function #f(x)=x^2#.

#F(x)=# #{{x^3}\over{3}}#

#\begin{array}{rcl}\displaystyle \int x^2 \; \dd x
&=&\displaystyle \frac{1}{2+1} x^{2+1}+C \\ &&\displaystyle \phantom{xxx}\blue{\text{rule of calculation } \int x^{n} \; \dd x = \displaystyle\frac{1}{n+1} x^{n+1} + C}\\
&=&\displaystyle {{x^3}\over{3}} +C \\ &&\phantom{xxx}\blue{\text{simplified}}
\end{array}#

Because we only asked for one antiderivative, we can now choose #C=0#. This gives:
\[F(x)={{x^3}\over{3}}\]

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