Calculus Examples

$f'' (x) = 32 x^{3} - 15 x^{2} + 8 x$

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.

$\int 32 x^{3} d x + \int - 15 x^{2} d x + \int 8 x d x$

Step 3

Since $32$ is constant with respect to $x$ , move $32$ out of the integral.

$32 \int x^{3} d x + \int - 15 x^{2} d x + \int 8 x d x$

Step 4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$32 (\frac{1}{4} x^{4} + C) + \int - 15 x^{2} d x + \int 8 x d x$

Step 5

Since $- 15$ is constant with respect to $x$ , move $- 15$ out of the integral.

$32 (\frac{1}{4} x^{4} + C) - 15 \int x^{2} d x + \int 8 x d x$

Step 6

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

$32 (\frac{1}{4} x^{4} + C) - 15 (\frac{1}{3} x^{3} + C) + \int 8 x d x$

Step 7

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

$32 (\frac{1}{4} x^{4} + C) - 15 (\frac{1}{3} x^{3} + C) + 8 \int x d x$

Step 8

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

$32 (\frac{1}{4} x^{4} + C) - 15 (\frac{1}{3} x^{3} + C) + 8 (\frac{1}{2} x^{2} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$8 x^{4} - 5 x^{3} + 8 (\frac{1}{2} x^{2}) + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$8 x^{4} - 5 x^{3} + 8 \frac{x^{2}}{2} + C$

Step 9.2.2

Combine $8$ and $\frac{x^{2}}{2}$ .

$8 x^{4} - 5 x^{3} + \frac{8 x^{2}}{2} + C$

Step 9.2.3

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

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Step 9.2.3.1

Factor $2$ out of $8 x^{2}$ .

$8 x^{4} - 5 x^{3} + \frac{2 (4 x^{2})}{2} + C$

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $2$ out of $2$ .

$8 x^{4} - 5 x^{3} + \frac{2 (4 x^{2})}{2 (1)} + C$

Step 9.2.3.2.2

Cancel the common factor.

$8 x^{4} - 5 x^{3} + \frac{2 (4 x^{2})}{2 \cdot 1} + C$

Step 9.2.3.2.3

Rewrite the expression.

$8 x^{4} - 5 x^{3} + \frac{4 x^{2}}{1} + C$

Step 9.2.3.2.4

Divide $4 x^{2}$ by $1$ .

$8 x^{4} - 5 x^{3} + 4 x^{2} + C$

Step 10

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

$f' (x) = 8 x^{4} - 5 x^{3} + 4 x^{2} + C$

Step 11

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 12

Split the single integral into multiple integrals.

$\int 8 x^{4} d x + \int - 5 x^{3} d x + \int 4 x^{2} d x + \int C d x$

Step 13

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

$8 \int x^{4} d x + \int - 5 x^{3} d x + \int 4 x^{2} d x + \int C d x$

Step 14

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$8 (\frac{1}{5} x^{5} + C) + \int - 5 x^{3} d x + \int 4 x^{2} d x + \int C d x$

Step 15

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$8 (\frac{1}{5} x^{5} + C) - 5 \int x^{3} d x + \int 4 x^{2} d x + \int C d x$

Step 16

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$8 (\frac{1}{5} x^{5} + C) - 5 (\frac{1}{4} x^{4} + C) + \int 4 x^{2} d x + \int C d x$

Step 17

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$8 (\frac{1}{5} x^{5} + C) - 5 (\frac{1}{4} x^{4} + C) + 4 \int x^{2} d x + \int C d x$

Step 18

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

$8 (\frac{1}{5} x^{5} + C) - 5 (\frac{1}{4} x^{4} + C) + 4 (\frac{1}{3} x^{3} + C) + \int C d x$

Step 19

Apply the constant rule.

$8 (\frac{1}{5} x^{5} + C) - 5 (\frac{1}{4} x^{4} + C) + 4 (\frac{1}{3} x^{3} + C) + C x + C$

Step 20

Simplify.

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Step 20.1

Simplify.

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Step 20.1.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$8 (\frac{x^{5}}{5} + C) - 5 (\frac{1}{4} x^{4} + C) + 4 (\frac{1}{3} x^{3} + C) + C x + C$

Step 20.1.2

Combine $\frac{1}{4}$ and $x^{4}$ .

$8 (\frac{x^{5}}{5} + C) - 5 (\frac{x^{4}}{4} + C) + 4 (\frac{1}{3} x^{3} + C) + C x + C$

Step 20.1.3

Combine $\frac{1}{3}$ and $x^{3}$ .

$8 (\frac{x^{5}}{5} + C) - 5 (\frac{x^{4}}{4} + C) + 4 (\frac{x^{3}}{3} + C) + C x + C$

Step 20.2

Simplify.

$\frac{8 x^{5}}{5} - \frac{5 x^{4}}{4} + \frac{4 x^{3}}{3} + C x + C$

Step 21

Reorder terms.

$\frac{8}{5} x^{5} - \frac{5}{4} x^{4} + \frac{4}{3} x^{3} + C x + C$

Step 22

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

$f (x) = \frac{8}{5} x^{5} - \frac{5}{4} x^{4} + \frac{4}{3} x^{3} + C x + C$