Calculus Examples

$(1 + 3 x) x^{2}$

Step 1

Write $(1 + 3 x) x^{2}$ as a function.

$f (x) = (1 + 3 x) x^{2}$

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.

$F (x) = \int (1 + 3 x) x^{2} d x$

Step 4

Expand $(1 + 3 x) x^{2}$ .

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

Raise $x$ to the power of $1$ .

$\int x^{2} + 3 (x^{1} x^{2}) d x$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.5

Add $1$ and $2$ .

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

Step 4.6

Reorder $x^{2}$ and $3 x^{3}$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$3 (\frac{1}{4} x^{4} + C) + \int x^{2} d x$

Step 8

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

$3 (\frac{1}{4} x^{4} + C) + \frac{1}{3} x^{3} + C$

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$3 (\frac{x^{4}}{4} + C) + \frac{1}{3} x^{3} + C$

Step 9.2

Simplify.

$\frac{3 x^{4}}{4} + \frac{1}{3} x^{3} + C$

Step 9.3

Reorder terms.

$\frac{3}{4} x^{4} + \frac{1}{3} x^{3} + C$

Step 10

The answer is the antiderivative of the function $f (x) = (1 + 3 x) x^{2}$ .

$F (x) =$ $\frac{3}{4} x^{4} + \frac{1}{3} x^{3} + C$