Calculus Examples

$f' (x) = 9 x^{2} - 8 x$

Step 1

The function $f (x)$ can be found by evaluating the indefinite integral of the derivative $f' (x)$ .

$f (x) = \int f' (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int 9 x^{2} d x + \int - 8 x d x$

Step 3

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int x^{2} d x + \int - 8 x d x$

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$9 (\frac{1}{3} x^{3} + C) + \int - 8 x d x$

Step 5

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$9 (\frac{1}{3} x^{3} + C) - 8 \int x d x$

Step 6

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$9 (\frac{1}{3} x^{3} + C) - 8 (\frac{1}{2} x^{2} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$9 (\frac{1}{3}) x^{3} - 8 (\frac{1}{2}) x^{2} + C$

Step 7.2

Simplify.

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Step 7.2.1

Combine $9$ and $\frac{1}{3}$ .

$\frac{9}{3} x^{3} - 8 (\frac{1}{2}) x^{2} + C$

Step 7.2.2

Cancel the common factor of $9$ and $3$ .

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Step 7.2.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 3}{3} x^{3} - 8 (\frac{1}{2}) x^{2} + C$

Step 7.2.2.2

Cancel the common factors.

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Step 7.2.2.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 3}{3 (1)} x^{3} - 8 (\frac{1}{2}) x^{2} + C$

Step 7.2.2.2.2

Cancel the common factor.

$\frac{3 \cdot 3}{3 \cdot 1} x^{3} - 8 (\frac{1}{2}) x^{2} + C$

Step 7.2.2.2.3

Rewrite the expression.

$\frac{3}{1} x^{3} - 8 (\frac{1}{2}) x^{2} + C$

Step 7.2.2.2.4

Divide $3$ by $1$ .

$3 x^{3} - 8 (\frac{1}{2}) x^{2} + C$

Step 7.2.3

Combine $- 8$ and $\frac{1}{2}$ .

$3 x^{3} + \frac{- 8}{2} x^{2} + C$

Step 7.2.4

Cancel the common factor of $- 8$ and $2$ .

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Step 7.2.4.1

Factor $2$ out of $- 8$ .

$3 x^{3} + \frac{2 \cdot - 4}{2} x^{2} + C$

Step 7.2.4.2

Cancel the common factors.

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Step 7.2.4.2.1

Factor $2$ out of $2$ .

$3 x^{3} + \frac{2 \cdot - 4}{2 (1)} x^{2} + C$

Step 7.2.4.2.2

Cancel the common factor.

$3 x^{3} + \frac{2 \cdot - 4}{2 \cdot 1} x^{2} + C$

Step 7.2.4.2.3

Rewrite the expression.

$3 x^{3} + \frac{- 4}{1} x^{2} + C$

Step 7.2.4.2.4

Divide $- 4$ by $1$ .

$3 x^{3} - 4 x^{2} + C$

Step 8

The function $f$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$f (x) = 3 x^{3} - 4 x^{2} + C$