Example 1: Antiderivative of a Constant

Definite Integral

Indefinite Integral

$\underset{3}{\overset{7}{\int }}5\mathit{ dx}$

$\int 5\mathit{ dx}$

Step 1: Simplify and look for algebraic rewrites.

None here.

$\underset{3}{\overset{7}{\int }}5\mathit{ dx}$

$\int 5\mathit{ dx}$

Step 2: Identify any term(s) that are just a constant.

The only term in either example is just a constant, 5.

$\underset{3}{\overset{7}{\int }}5\mathit{ dx}$

$\int 5\mathit{ dx}$

Step 3: Take the antiderivativeof the constant using the Recipe: Numbers by themselves, constants, gain a letter back.

$\underset{3}{\overset{7}{\int }}5\mathit{ dx}\mathit{=}{5x|}_{x=3}^{x=7}$

$\int 5\mathit{ dx}\mathit{=}5x\mathit{+}\mathit{C}$

DO NOT FORGET THE +C

Step 4 ( Definite Integral ONLY ): Evaluate the antiderivative result using the Top – Bottom method.

$\underset{3}{\overset{7}{\int }}5\mathit{ dx}\mathit{=}{5x|}_{x=3}^{x=7}\mathit{=}5\left(7\right)–5\left(3\right)=20$

Final Result Meaning: Remember the Definite Integral will always provide you a definite value , and the Indefinite Integral provides you a family of solutions .

The Net Area between the curve $f\left(x\right)=5$ and the x-axis on the x-interval [ 3 , 7 ] is 20.

Since the final result is positive, you know without even seeing the graph that there is more area above the x-axis than below it.

The antiderivative of the equation $f‘\left(x\right)=5$is the family of graphs $f\left(x\right)=5x\mathit{+}\mathit{C}$.

All graphs of the form $f\left(x\right)=5x\mathit{+}\mathit{C}$ have the same derivative (instantaneous rate of change), $f‘\left(x\right)=5$, at any x-value .

The only difference between any of original graphs, $f\left(x\right)=5x\mathit{+}\mathit{C}$, is just a vertical shift of +C .

In the graph above are examples of 3-different C-values. You can see the instantaneous rate of change ( slope ) is the same at every x-value between all the different examples, no matter the +C .