Calculus Examples

$\frac{6 x^{3} - 3 x + 2}{5}$

Step 1

Write $\frac{6 x^{3} - 3 x + 2}{5}$ as a function.

$f (x) = \frac{6 x^{3} - 3 x + 2}{5}$

Step 2

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

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

Step 3

Set up the integral to solve.

$F (x) = \int \frac{6 x^{3} - 3 x + 2}{5} d x$

Step 4

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{5} \int 6 x^{3} - 3 x + 2 d x$

Step 5

Split the single integral into multiple integrals.

$\frac{1}{5} (\int 6 x^{3} d x + \int - 3 x d x + \int 2 d x)$

Step 6

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\frac{1}{5} (6 \int x^{3} d x + \int - 3 x d x + \int 2 d x)$

Step 7

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

$\frac{1}{5} (6 (\frac{1}{4} x^{4} + C) + \int - 3 x d x + \int 2 d x)$

Step 8

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$\frac{1}{5} (6 (\frac{1}{4} x^{4} + C) - 3 \int x d x + \int 2 d x)$

Step 9

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

$\frac{1}{5} (6 (\frac{1}{4} x^{4} + C) - 3 (\frac{1}{2} x^{2} + C) + \int 2 d x)$

Step 10

Apply the constant rule.

$\frac{1}{5} (6 (\frac{1}{4} x^{4} + C) - 3 (\frac{1}{2} x^{2} + C) + 2 x + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

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Step 11.1.1

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

$\frac{1}{5} (6 (\frac{x^{4}}{4} + C) - 3 (\frac{1}{2} x^{2} + C) + 2 x + C)$

Step 11.1.2

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

$\frac{1}{5} (6 (\frac{x^{4}}{4} + C) - 3 (\frac{x^{2}}{2} + C) + 2 x + C)$

Step 11.2

Simplify.

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

Step 12

Reorder terms.

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

Step 13

The answer is the antiderivative of the function $f (x) = \frac{6 x^{3} - 3 x + 2}{5}$ .

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