Calculus Examples

$2 x^{5}$

Step 1

Write $2 x^{5}$ as a function.

$f (x) = 2 x^{5}$

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.

$F (x) = \int 2 x^{5} d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x^{5} d x$

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Rewrite $2 (\frac{1}{6} x^{6} + C)$ as $2 (\frac{1}{6}) x^{6} + C$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Combine $2$ and $\frac{1}{6}$ .

$\frac{2}{6} x^{6} + C$

Step 6.2.2

Cancel the common factor of $2$ and $6$ .

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Step 6.2.2.1

Factor $2$ out of $2$ .

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

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $2$ out of $6$ .

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

Step 6.2.2.2.2

Cancel the common factor.

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

Step 6.2.2.2.3

Rewrite the expression.

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

Step 7

The answer is the antiderivative of the function $f (x) = 2 x^{5}$ .

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