Calculus Examples

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

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

$\frac{1}{3} x^{3} + C + \int 2 x d x$

Step 5

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

$\frac{1}{3} x^{3} + C + 2 \int x d x$

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Simplify.

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

Cancel the common factor of $2$ .

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Step 7.2.2.1

Cancel the common factor.

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

Step 7.2.2.2

Rewrite the expression.

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

Step 7.2.3

Multiply $x^{2}$ by $1$ .

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

Step 8

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

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