Constant of Integration
Simple overview of the constant of integration.
Constant of Integration Motivation
What happens we we find the following three derivatives?
\[\frac{d}{dx}\left(x^4+3\right),\qquad \frac{d}{dx}\left(x^4+20\right),\qquad \frac{d}{dx}\left(x^4-20\right)\]The power rule gives us
\[\frac{d}{dx}\left(x^4\right)=4x^3\]Now since the derivative of a constant is zero,
\[\frac{d}{dx}(3)=0,\qquad \frac{d}{dx}(20)=0,\qquad \frac{d}{dx}(-20)=0\]we end up getting the same derivative for all three examples
\[\frac{d}{dx}\left(x^4+3\right)=4x^3\] \[\frac{d}{dx}\left(x^4+20\right)=4x^3\] \[\frac{d}{dx}\left(x^4-20\right)=4x^3\]What happens when we revert the process?
Watch this section: derivative demonstration at 0:21.
Antiderivatives Contain no Knowledge of Constant Lost Through Differentiation
If the derivative of all three original examples produced $4x^3$, then integrating $4x^3$ could potentially map back to any three of the starting examples.
The antiderivative begins with the power rule for integration
\[\int 4x^3 dx=4\cdot\frac{x^4}{4}\]After simplifying
\[4\cdot\frac{x^4}{4}=x^4\]But $x^4$ alone does not tell us which one of $x^4+3$, $x^4+20$, $x^4-20$, or another function with a different constant is appropriate.
Since we cannot confidently pick the correct constant (without some initial conditions) we use the “$+C$” to represent all possible scenarios!
\[\int 4x^3 dx=x^4+C\]Here, $C$ represents the constant that may have disappeared during differentiation when we take antiderivatives.
Watch this section: going back to the antiderivative at 1:21.
Timestamp Guide
Related Calculus Work
This video uses the power rule in both directions. For derivative-side practice, use the differentiation power rule examples.
For another integration companion page, review u-substitution notes. The video description also points students toward the JoeCMath integration playlist: watch the integration playlist on YouTube.