Example 2: Indefinite Integral Power Rule

$\int x^3+{5x}^2–7x+3\mathit{ dx}$

Step 1: Simplify and look for algebraic rewrites.

When looking at the first chunk Remind yourself that there is really an unwritten 1 for the power. That is what allows us to apply the power rule .

$\int x^3+{5x}^2–7x^1+3\mathit{ dx}$

Step 2: Identify any term(s) that include variables raised to a power.

Break the problem down into bitesize chunks based upon the + and – , and identify the antiderivative rule for each chunk (term).

Here you have 4 chunks. The first three chunks will be a power rule , and the final chunk is a constant.

$\int x^3+{5x}^2–7x^1+3\mathit{ dx}$

Step 3: Take the antiderivativeof the variables raised to a power using the Recipe:Add 1 to the power; Divide by the new power.

Chunk 1: Power Rule

Chunk 2: Power Rule

Chunk 3: Power Rule

Chunk 4: Constant Rule

You might find it helpful (I do) to slide your fractions to the front of each term.

$\int x^3+{5x}^2–7x^1+3\mathit{ }\mathit{dx}$

$=\frac{x^{3+1}}{4}+\frac{5x^{2+1}}{3}–\frac{7x^{1+1}}{2}+3x\mathit{+}\mathit{C}$

$=\frac{x^4}{4}+\frac{5x^3}{3}–\frac{7x^2}{2}+3x\mathit{+}\mathit{C}$

$=\frac{1}{4}{x}^4+\frac{5}{3}{x}^3–\frac{7}{2}{x}^2+3x\mathit{+}\mathit{C}$

DO NOT FORGET THE +C

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 antiderivative of the equation $f\prime \left(x\right)=x^3+{5x}^2–7x+3$is the family of graphs $f\left(x\right)=\frac{1}{4}{x}^4+\frac{5}{3}{x}^3–\frac{7}{2}{x}^2+3x\mathit{+}\mathit{C}$.

All graphs of the form $f\left(x\right)=\frac{1}{4}{x}^4+\frac{5}{3}{x}^3–\frac{7}{2}{x}^2+3x\mathit{+}\mathit{C}$ have the same derivative (instantaneous rate of change), $f\prime \left(x\right)=x^3+{5x}^2–7x+3$, at any x-value .

The only difference between any of original graphs, $f\left(x\right)=\frac{1}{4}{x}^4+\frac{5}{3}{x}^3–\frac{7}{2}{x}^2+3x\mathit{+}\mathit{C}$, 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 .