Calculus Examples

$f (x) = 3 x^{2} + p x$

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.

$F (x) = \int 3 x^{2} + p x d x$

Step 3

Split the single integral into multiple integrals.

$\int 3 x^{2} d x + \int p x d x$

Step 4

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

$3 \int x^{2} d x + \int p x d x$

Step 5

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

$3 (\frac{1}{3} x^{3} + C) + \int p x d x$

Step 6

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

$3 (\frac{1}{3} x^{3} + C) + p \int x d x$

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

Combine $3$ and $\frac{1}{3}$ .

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

Step 8.2.2

Cancel the common factor of $3$ .

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Step 8.2.2.1

Cancel the common factor.

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

Step 8.2.2.2

Rewrite the expression.

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

Step 8.2.3

Multiply $x^{3}$ by $1$ .

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

Step 8.2.4

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

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

Step 8.2.5

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

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

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

Step 9

The answer is the antiderivative of the function $f (x) = 3 x^{2} + p x$ .

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