Calculus Examples

$6 x - 9 + 3 x^{5}$

Step 1

Write $6 x - 9 + 3 x^{5}$ as a function.

$f (x) = 6 x - 9 + 3 x^{5}$

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.

$F (x) = \int 6 x - 9 + 3 x^{5} d x$

Step 4

Split the single integral into multiple integrals.

$\int 6 x d x + \int - 9 d x + \int 3 x^{5} d x$

Step 5

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

$6 \int x d x + \int - 9 d x + \int 3 x^{5} d x$

Step 6

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

$6 (\frac{1}{2} x^{2} + C) + \int - 9 d x + \int 3 x^{5} d x$

Step 7

Apply the constant rule.

$6 (\frac{1}{2} x^{2} + C) - 9 x + C + \int 3 x^{5} d x$

Step 8

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

$6 (\frac{x^{2}}{2} + C) - 9 x + C + \int 3 x^{5} d x$

Step 9

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

$6 (\frac{x^{2}}{2} + C) - 9 x + C + 3 \int x^{5} d x$

Step 10

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$6 (\frac{x^{2}}{2} + C) - 9 x + C + 3 (\frac{1}{6} x^{6} + C)$

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{6}$ and $x^{6}$ .

$6 (\frac{x^{2}}{2} + C) - 9 x + C + 3 (\frac{x^{6}}{6} + C)$

Step 11.2

Simplify.

$3 x^{2} - 9 x + 3 \frac{x^{6}}{6} + C$

Step 11.3

Simplify.

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Step 11.3.1

Combine $3$ and $\frac{x^{6}}{6}$ .

$3 x^{2} - 9 x + \frac{3 x^{6}}{6} + C$

Step 11.3.2

Cancel the common factor of $3$ and $6$ .

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Step 11.3.2.1

Factor $3$ out of $3 x^{6}$ .

$3 x^{2} - 9 x + \frac{3 (x^{6})}{6} + C$

Step 11.3.2.2

Cancel the common factors.

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Step 11.3.2.2.1

Factor $3$ out of $6$ .

$3 x^{2} - 9 x + \frac{3 x^{6}}{3 \cdot 2} + C$

Step 11.3.2.2.2

Cancel the common factor.

$3 x^{2} - 9 x + \frac{3 x^{6}}{3 \cdot 2} + C$

Step 11.3.2.2.3

Rewrite the expression.

$3 x^{2} - 9 x + \frac{x^{6}}{2} + C$

Step 11.4

Reorder terms.

$3 x^{2} - 9 x + \frac{1}{2} x^{6} + C$

Step 12

The answer is the antiderivative of the function $f (x) = 6 x - 9 + 3 x^{5}$ .

$F (x) =$ $3 x^{2} - 9 x + \frac{1}{2} x^{6} + C$