Calculus Examples

$\int_{0}^{5} 6 \sqrt{x + 4} d x$

Step 1

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

$6 \int_{0}^{5} \sqrt{x + 4} d x$

Step 2

Let $u = x + 4$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.1

Let $u = x + 4$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $x + 4$ .

$\frac{d}{d x} [x + 4]$

Step 2.1.2

By the Sum Rule, the derivative of $x + 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [4]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [4]$

Step 2.1.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

Substitute the lower limit in for $x$ in $u = x + 4$ .

$u_{lower} = 0 + 4$

Step 2.3

Add $0$ and $4$ .

$u_{lower} = 4$

Step 2.4

Substitute the upper limit in for $x$ in $u = x + 4$ .

$u_{upper} = 5 + 4$

Step 2.5

Add $5$ and $4$ .

$u_{upper} = 9$

Step 2.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 4$

$u_{upper} = 9$

Step 2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$6 \int_{4}^{9} \sqrt{u} d u$

Step 3

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

$6 \int_{4}^{9} u^{\frac{1}{2}} d u$

Step 4

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

$6 (\frac{2}{3} u^{\frac{3}{2}}]_{4}^{9})$

Step 5

Combine $\frac{2}{3}$ and $u^{\frac{3}{2}}$ .

$6 (\frac{2 u^{\frac{3}{2}}}{3}]_{4}^{9})$

Step 6

Substitute and simplify.

Tap for more steps...

Step 6.1

Evaluate $\frac{2 u^{\frac{3}{2}}}{3}$ at $9$ and at $4$ .

$6 ((\frac{2 \cdot 9^{\frac{3}{2}}}{3}) - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

Rewrite $9$ as $3^{2}$ .

$6 (\frac{2 \cdot {(3^{2})}^{\frac{3}{2}}}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2.2

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

$6 (\frac{2 \cdot 3^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.3.1

Cancel the common factor.

$6 (\frac{2 \cdot 3^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2.3.2

Rewrite the expression.

$6 (\frac{2 \cdot 3^{3}}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2.4

Raise $3$ to the power of $3$ .

$6 (\frac{2 \cdot 27}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2.5

Multiply $2$ by $27$ .

$6 (\frac{54}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2.6

Cancel the common factor of $54$ and $3$ .

Tap for more steps...

Step 6.2.6.1

Factor $3$ out of $54$ .

$6 (\frac{3 \cdot 18}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2.6.2

Cancel the common factors.

Tap for more steps...

Step 6.2.6.2.1

Factor $3$ out of $3$ .

$6 (\frac{3 \cdot 18}{3 (1)} - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2.6.2.2

Cancel the common factor.

$6 (\frac{3 \cdot 18}{3 \cdot 1} - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2.6.2.3

Rewrite the expression.

$6 (\frac{18}{1} - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2.6.2.4

Divide $18$ by $1$ .

$6 (18 - \frac{2 \cdot 4^{\frac{3}{2}}}{3})$

Step 6.2.7

Rewrite $4$ as $2^{2}$ .

$6 (18 - \frac{2 \cdot {(2^{2})}^{\frac{3}{2}}}{3})$

Step 6.2.8

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

$6 (18 - \frac{2 \cdot 2^{2 (\frac{3}{2})}}{3})$

Step 6.2.9

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.9.1

Cancel the common factor.

$6 (18 - \frac{2 \cdot 2^{2 (\frac{3}{2})}}{3})$

Step 6.2.9.2

Rewrite the expression.

$6 (18 - \frac{2 \cdot 2^{3}}{3})$

Step 6.2.10

Raise $2$ to the power of $3$ .

$6 (18 - \frac{2 \cdot 8}{3})$

Step 6.2.11

Multiply $2$ by $8$ .

$6 (18 - \frac{16}{3})$

Step 6.2.12

To write $18$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6 (18 \cdot \frac{3}{3} - \frac{16}{3})$

Step 6.2.13

Combine $18$ and $\frac{3}{3}$ .

$6 (\frac{18 \cdot 3}{3} - \frac{16}{3})$

Step 6.2.14

Combine the numerators over the common denominator.

$6 \frac{18 \cdot 3 - 16}{3}$

Step 6.2.15

Simplify the numerator.

Tap for more steps...

Step 6.2.15.1

Multiply $18$ by $3$ .

$6 \frac{54 - 16}{3}$

Step 6.2.15.2

Subtract $16$ from $54$ .

$6 (\frac{38}{3})$

Step 6.2.16

Combine $6$ and $\frac{38}{3}$ .

$\frac{6 \cdot 38}{3}$

Step 6.2.17

Multiply $6$ by $38$ .

$\frac{228}{3}$

Step 6.2.18

Cancel the common factor of $228$ and $3$ .

Tap for more steps...

Step 6.2.18.1

Factor $3$ out of $228$ .

$\frac{3 \cdot 76}{3}$

Step 6.2.18.2

Cancel the common factors.

Tap for more steps...

Step 6.2.18.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 76}{3 (1)}$

Step 6.2.18.2.2

Cancel the common factor.

$\frac{3 \cdot 76}{3 \cdot 1}$

Step 6.2.18.2.3

Rewrite the expression.

$\frac{76}{1}$

Step 6.2.18.2.4

Divide $76$ by $1$ .

$76$

Step 7