Calculus Examples

$f (x) = 5 \cdot 2 x$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int 5 \cdot (2 x) d x$

Step 3

Multiply $2$ by $5$ .

$\int 10 x d x$

Step 4

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$10 \int x d x$

Step 5

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

$10 (\frac{1}{2} x^{2} + C)$

Step 6

Simplify the answer.

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Step 6.1

Rewrite $10 (\frac{1}{2} x^{2} + C)$ as $10 (\frac{1}{2}) x^{2} + C$ .

$10 (\frac{1}{2}) x^{2} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine $10$ and $\frac{1}{2}$ .

$\frac{10}{2} x^{2} + C$

Step 6.2.2

Cancel the common factor of $10$ and $2$ .

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Step 6.2.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 5}{2} x^{2} + C$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 5}{2 (1)} x^{2} + C$

Step 6.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 5}{2 \cdot 1} x^{2} + C$

Step 6.2.2.2.3

Rewrite the expression.

$\frac{5}{1} x^{2} + C$

Step 6.2.2.2.4

Divide $5$ by $1$ .

$5 x^{2} + C$

Step 7

The answer is the antiderivative of the function $f (x) = 5 \cdot (2 x)$ .

$F (x) =$ $5 x^{2} + C$