Calculus Examples

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

Step 1

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

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

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.

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

Step 4

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

Tap for more steps...

Step 4.1

Rewrite ${(3 + x)}^{2}$ as $(3 + x) (3 + x)$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Apply the distributive property.

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

Step 4.6

Apply the distributive property.

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

Step 4.7

Apply the distributive property.

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

Step 4.8

Move $x^{2}$ .

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

Step 4.9

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

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

Step 4.10

Reorder $x$ and $3$ .

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

Step 4.11

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

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

Step 4.12

Multiply $3$ by $3$ .

$\int 9 x^{2} + 3 x^{2} x + 3 x^{2} x + x^{2} x \cdot x d x$

Step 4.13

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

$\int 9 x^{2} + 3 (x^{2} x^{1}) + 3 x^{2} x + x^{2} x \cdot x d x$

Step 4.14

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

$\int 9 x^{2} + 3 x^{2 + 1} + 3 x^{2} x + x^{2} x \cdot x d x$

Step 4.15

Add $2$ and $1$ .

$\int 9 x^{2} + 3 x^{3} + 3 x^{2} x + x^{2} x \cdot x d x$

Step 4.16

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

$\int 9 x^{2} + 3 x^{3} + 3 (x^{2} x^{1}) + x^{2} x \cdot x d x$

Step 4.17

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

$\int 9 x^{2} + 3 x^{3} + 3 x^{2 + 1} + x^{2} x \cdot x d x$

Step 4.18

Add $2$ and $1$ .

$\int 9 x^{2} + 3 x^{3} + 3 x^{3} + x^{2} x \cdot x d x$

Step 4.19

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

$\int 9 x^{2} + 3 x^{3} + 3 x^{3} + x^{2} x^{1} x d x$

Step 4.20

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

$\int 9 x^{2} + 3 x^{3} + 3 x^{3} + x^{2 + 1} x d x$

Step 4.21

Add $2$ and $1$ .

$\int 9 x^{2} + 3 x^{3} + 3 x^{3} + x^{3} x d x$

Step 4.22

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

$\int 9 x^{2} + 3 x^{3} + 3 x^{3} + x^{3} x^{1} d x$

Step 4.23

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

$\int 9 x^{2} + 3 x^{3} + 3 x^{3} + x^{3 + 1} d x$

Step 4.24

Add $3$ and $1$ .

$\int 9 x^{2} + 3 x^{3} + 3 x^{3} + x^{4} d x$

Step 4.25

Add $3 x^{3}$ and $3 x^{3}$ .

$\int 9 x^{2} + 6 x^{3} + x^{4} d x$

Step 4.26

Reorder $6 x^{3}$ and $x^{4}$ .

$\int 9 x^{2} + x^{4} + 6 x^{3} d x$

Step 4.27

Move $9 x^{2}$ .

$\int x^{4} + 6 x^{3} + 9 x^{2} d x$

$\int x^{4} + 6 x^{3} + 9 x^{2} d x$

Step 5

Split the single integral into multiple integrals.

$\int x^{4} d x + \int 6 x^{3} d x + \int 9 x^{2} d x$

Step 6

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{5} x^{5} + C + \int 6 x^{3} d x + \int 9 x^{2} d x$

Step 7

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\frac{1}{5} x^{5} + C + 6 \int x^{3} d x + \int 9 x^{2} d x$

Step 8

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

$\frac{1}{5} x^{5} + C + 6 (\frac{1}{4} x^{4} + C) + \int 9 x^{2} d x$

Step 9

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$\frac{1}{5} x^{5} + C + 6 (\frac{1}{4} x^{4} + C) + 9 \int x^{2} d x$

Step 10

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

$\frac{1}{5} x^{5} + C + 6 (\frac{1}{4} x^{4} + C) + 9 (\frac{1}{3} x^{3} + C)$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Simplify.

$\frac{x^{5}}{5} + \frac{3 x^{4}}{2} + 9 (\frac{1}{3} x^{3}) + C$

Step 11.2

Simplify.

Tap for more steps...

Step 11.2.1

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

$\frac{x^{5}}{5} + \frac{3 x^{4}}{2} + 9 \frac{x^{3}}{3} + C$

Step 11.2.2

Combine $9$ and $\frac{x^{3}}{3}$ .

$\frac{x^{5}}{5} + \frac{3 x^{4}}{2} + \frac{9 x^{3}}{3} + C$

Step 11.2.3

Cancel the common factor of $9$ and $3$ .

Tap for more steps...

Step 11.2.3.1

Factor $3$ out of $9 x^{3}$ .

$\frac{x^{5}}{5} + \frac{3 x^{4}}{2} + \frac{3 (3 x^{3})}{3} + C$

Step 11.2.3.2

Cancel the common factors.

Tap for more steps...

Step 11.2.3.2.1

Factor $3$ out of $3$ .

$\frac{x^{5}}{5} + \frac{3 x^{4}}{2} + \frac{3 (3 x^{3})}{3 (1)} + C$

Step 11.2.3.2.2

Cancel the common factor.

$\frac{x^{5}}{5} + \frac{3 x^{4}}{2} + \frac{3 (3 x^{3})}{3 \cdot 1} + C$

Step 11.2.3.2.3

Rewrite the expression.

$\frac{x^{5}}{5} + \frac{3 x^{4}}{2} + \frac{3 x^{3}}{1} + C$

Step 11.2.3.2.4

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

$\frac{x^{5}}{5} + \frac{3 x^{4}}{2} + 3 x^{3} + C$

$\frac{x^{5}}{5} + \frac{3 x^{4}}{2} + 3 x^{3} + C$

$\frac{x^{5}}{5} + \frac{3 x^{4}}{2} + 3 x^{3} + C$

$\frac{x^{5}}{5} + \frac{3 x^{4}}{2} + 3 x^{3} + C$

Step 11.3

Reorder terms.

$\frac{1}{5} x^{5} + \frac{3}{2} x^{4} + 3 x^{3} + C$

$\frac{1}{5} x^{5} + \frac{3}{2} x^{4} + 3 x^{3} + C$

Step 12

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

$F (x) =$ $\frac{1}{5} x^{5} + \frac{3}{2} x^{4} + 3 x^{3} + C$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$