Constant of Integration
This page follows the JoeCMath video +C, The Constant of Integration. The video explains why an indefinite integral ends with $+C$: derivatives erase constants, so an antiderivative has to leave room for any constant that could have been present before differentiating.
Main Idea
An indefinite integral asks for a family of antiderivatives. The $+C$ is the way we show that the answer could differ by a constant.
A compact way to say the idea is:
\[\int f'(x)\,dx=f(x)+C.\]The video does not introduce $+C$ as a formality. It builds the idea by showing what happens when several functions have the same derivative.
Watch this section: introduction at 0:00.
Constants Disappear Under Derivatives
The video starts by differentiating three functions that only differ by a constant:
\[\frac{d}{dx}\left(x^4+3\right),\qquad \frac{d}{dx}\left(x^4+20\right),\qquad \frac{d}{dx}\left(x^4-20\right).\]The power rule gives
\[\frac{d}{dx}\left(x^4\right)=4x^3.\]The constants all become zero:
\[\frac{d}{dx}(3)=0,\qquad \frac{d}{dx}(20)=0,\qquad \frac{d}{dx}(-20)=0.\]So all three derivatives have the same result:
\[\frac{d}{dx}\left(x^4+3\right)=4x^3,\] \[\frac{d}{dx}\left(x^4+20\right)=4x^3,\]and
\[\frac{d}{dx}\left(x^4-20\right)=4x^3.\]That is the reason $+C$ is needed later. Once the derivative has been taken, the original constant is no longer visible.
Watch this section: derivative demonstration at 0:21.
Going Backward Needs C
After the constants disappear, the video turns the process around. If all three original functions produced $4x^3$, then integrating $4x^3$ should not pretend there was only one possible starting function.
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 whether the original function was $x^4+3$, $x^4+20$, $x^4-20$, or another function with a different constant. The indefinite integral keeps all of those possibilities by writing
\[\int 4x^3\,dx=x^4+C.\]Here, $C$ represents the constant that may have disappeared during differentiation.
Watch this section: going back to the antiderivative at 1:21.
Finding A Specific Value Of C
The second part of the video shows how $C$ changes from an unknown constant into a specific number.
The setup is:
\[F(x)=\int 3x^2\,dx\]with the extra information
\[F(1)=4.\]Start by finding the general antiderivative:
\[F(x)=\int 3x^2\,dx.\]Using the integration power rule,
\[F(x)=3\cdot\frac{x^3}{3}+C.\]The threes cancel, so
\[F(x)=x^3+C.\]At this point, the integral has given the whole family of possible answers. The condition $F(1)=4$ tells us which member of that family we need.
Watch this section: initial value setup at 2:23.
Using The Initial Value
To use $F(1)=4$, plug $1$ into the formula for $F(x)$:
\[F(1)=1^3+C.\]Since the video tells us $F(1)=4$, replace the left side with $4$:
\[4=1^3+C.\]Then simplify:
\[4=1+C.\]Subtract $1$ from both sides:
\[3=C.\]Now the constant is known, so the specific function is
\[F(x)=x^3+3.\]The important difference is that the indefinite integral first gives $x^3+C$, and the initial value decides which constant belongs in the final answer.
Watch this section: plugging in $F(1)=4$ at 3:22.
Recap From The Video
The video closes with the same core idea: add $C$ after an indefinite integral because a constant could have been lost when differentiating.
If no initial value is given, leave the answer with $+C$:
\[\int f'(x)\,dx=f(x)+C.\]If an initial value is given, use it to solve for $C$ and write the particular antiderivative.
Watch this section: recap at 4:11.
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.