Calculus Examples

$\int x {(6 + \sqrt{x})}^{2} d x$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Rewrite ${(6 + \sqrt{x})}^{2}$ as $(6 + \sqrt{x}) (6 + \sqrt{x})$ .

$\int x ((6 + \sqrt{x}) (6 + \sqrt{x})) d x$

Step 1.2

Expand $(6 + \sqrt{x}) (6 + \sqrt{x})$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\int x (6 (6 + \sqrt{x}) + \sqrt{x} (6 + \sqrt{x})) d x$

Step 1.2.2

Apply the distributive property.

$\int x (6 \cdot 6 + 6 \sqrt{x} + \sqrt{x} (6 + \sqrt{x})) d x$

Step 1.2.3

Apply the distributive property.

$\int x (6 \cdot 6 + 6 \sqrt{x} + \sqrt{x} \cdot 6 + \sqrt{x} \sqrt{x}) d x$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Multiply $6$ by $6$ .

$\int x (36 + 6 \sqrt{x} + \sqrt{x} \cdot 6 + \sqrt{x} \sqrt{x}) d x$

Step 1.3.1.2

Move $6$ to the left of $\sqrt{x}$ .

$\int x (36 + 6 \sqrt{x} + 6 \cdot \sqrt{x} + \sqrt{x} \sqrt{x}) d x$

Step 1.3.1.3

Multiply $\sqrt{x} \sqrt{x}$ .

Tap for more steps...

Step 1.3.1.3.1

Raise $\sqrt{x}$ to the power of $1$ .

$\int x (36 + 6 \sqrt{x} + 6 \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x}) d x$

Step 1.3.1.3.2

Raise $\sqrt{x}$ to the power of $1$ .

$\int x (36 + 6 \sqrt{x} + 6 \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1}) d x$

Step 1.3.1.3.3

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

$\int x (36 + 6 \sqrt{x} + 6 \sqrt{x} + {\sqrt{x}}^{1 + 1}) d x$

Step 1.3.1.3.4

Add $1$ and $1$ .

$\int x (36 + 6 \sqrt{x} + 6 \sqrt{x} + {\sqrt{x}}^{2}) d x$

Step 1.3.1.4

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

Tap for more steps...

Step 1.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int x (36 + 6 \sqrt{x} + 6 \sqrt{x} + {(x^{\frac{1}{2}})}^{2}) d x$

Step 1.3.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x (36 + 6 \sqrt{x} + 6 \sqrt{x} + x^{\frac{1}{2} \cdot 2}) d x$

Step 1.3.1.4.3

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

$\int x (36 + 6 \sqrt{x} + 6 \sqrt{x} + x^{\frac{2}{2}}) d x$

Step 1.3.1.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.3.1.4.4.1

Cancel the common factor.

$\int x (36 + 6 \sqrt{x} + 6 \sqrt{x} + x^{\frac{2}{2}}) d x$

Step 1.3.1.4.4.2

Rewrite the expression.

$\int x (36 + 6 \sqrt{x} + 6 \sqrt{x} + x^{1}) d x$

Step 1.3.1.4.5

Simplify.

$\int x (36 + 6 \sqrt{x} + 6 \sqrt{x} + x) d x$

Step 1.3.2

Add $6 \sqrt{x}$ and $6 \sqrt{x}$ .

$\int x (36 + 12 \sqrt{x} + x) d x$

Step 1.4

Apply the distributive property.

$\int x \cdot 36 + x (12 \sqrt{x}) + x \cdot x d x$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Move $36$ to the left of $x$ .

$\int 36 \cdot x + x (12 \sqrt{x}) + x \cdot x d x$

Step 1.5.2

Rewrite using the commutative property of multiplication.

$\int 36 \cdot x + 12 x \sqrt{x} + x \cdot x d x$

Step 1.5.3

Multiply $x$ by $x$ .

$\int 36 \cdot x + 12 x \sqrt{x} + x^{2} d x$

$\int 36 x + 12 x \sqrt{x} + x^{2} d x$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int 36 x + 12 x \cdot x^{\frac{1}{2}} + x^{2} d x$

Step 2.2

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 2.2.1

Move $x^{\frac{1}{2}}$ .

$\int 36 x + 12 (x^{\frac{1}{2}} x) + x^{2} d x$

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

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

$\int 36 x + 12 (x^{\frac{1}{2}} x^{1}) + x^{2} d x$

Step 2.2.2.2

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

$\int 36 x + 12 x^{\frac{1}{2} + 1} + x^{2} d x$

Step 2.2.3

Write $1$ as a fraction with a common denominator.

$\int 36 x + 12 x^{\frac{1}{2} + \frac{2}{2}} + x^{2} d x$

Step 2.2.4

Combine the numerators over the common denominator.

$\int 36 x + 12 x^{\frac{1 + 2}{2}} + x^{2} d x$

Step 2.2.5

Add $1$ and $2$ .

$\int 36 x + 12 x^{\frac{3}{2}} + x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$\int 36 x d x + \int 12 x^{\frac{3}{2}} d x + \int x^{2} d x$

Step 4

Since $36$ is constant with respect to $x$ , move $36$ out of the integral.

$36 \int x d x + \int 12 x^{\frac{3}{2}} d x + \int x^{2} d x$

Step 5

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

$36 (\frac{1}{2} x^{2} + C) + \int 12 x^{\frac{3}{2}} d x + \int x^{2} d x$

Step 6

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$36 (\frac{1}{2} x^{2} + C) + 12 \int x^{\frac{3}{2}} d x + \int x^{2} d x$

Step 7

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$36 (\frac{1}{2} x^{2} + C) + 12 (\frac{2}{5} x^{\frac{5}{2}} + C) + \int x^{2} d x$

Step 8

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

$36 (\frac{1}{2} x^{2} + C) + 12 (\frac{2}{5} x^{\frac{5}{2}} + C) + \frac{1}{3} x^{3} + C$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Simplify.

Tap for more steps...

Step 9.1.1

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

$36 (\frac{x^{2}}{2} + C) + 12 (\frac{2}{5} x^{\frac{5}{2}} + C) + \frac{1}{3} x^{3} + C$

Step 9.1.2

Combine $\frac{2}{5}$ and $x^{\frac{5}{2}}$ .

$36 (\frac{x^{2}}{2} + C) + 12 (\frac{2 x^{\frac{5}{2}}}{5} + C) + \frac{1}{3} x^{3} + C$

Step 9.2

Simplify.

$18 x^{2} + \frac{24 x^{\frac{5}{2}}}{5} + \frac{1}{3} x^{3} + C$

Step 9.3

Reorder terms.

$18 x^{2} + \frac{24}{5} x^{\frac{5}{2}} + \frac{1}{3} x^{3} + C$