Calculus Examples

$\int_{24}^{39} 2 p (10 \sqrt{x}) \sqrt{1 + {(\frac{5}{\sqrt{x}})}^{2}} d x$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Multiply $10$ by $2$ .

$\int_{24}^{39} 20 p \sqrt{x} \sqrt{1 + {(\frac{5}{\sqrt{x}})}^{2}} d x$

Step 1.2

Combine using the product rule for radicals.

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5}{\sqrt{x}})}^{2}) x} d x$

Step 1.3

Multiply $\frac{5}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}})}^{2}) x} d x$

Step 1.4

Combine and simplify the denominator.

Tap for more steps...

Step 1.4.1

Multiply $\frac{5}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5 \sqrt{x}}{\sqrt{x} \sqrt{x}})}^{2}) x} d x$

Step 1.4.2

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

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5 \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}})}^{2}) x} d x$

Step 1.4.3

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

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5 \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}})}^{2}) x} d x$

Step 1.4.4

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

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5 \sqrt{x}}{{\sqrt{x}}^{1 + 1}})}^{2}) x} d x$

Step 1.4.5

Add $1$ and $1$ .

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5 \sqrt{x}}{{\sqrt{x}}^{2}})}^{2}) x} d x$

Step 1.4.6

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

Tap for more steps...

Step 1.4.6.1

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

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5 \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}})}^{2}) x} d x$

Step 1.4.6.2

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

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5 \sqrt{x}}{x^{\frac{1}{2} \cdot 2}})}^{2}) x} d x$

Step 1.4.6.3

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

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5 \sqrt{x}}{x^{\frac{2}{2}}})}^{2}) x} d x$

Step 1.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.6.4.1

Cancel the common factor.

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5 \sqrt{x}}{x^{\frac{2}{2}}})}^{2}) x} d x$

Step 1.4.6.4.2

Rewrite the expression.

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5 \sqrt{x}}{x^{1}})}^{2}) x} d x$

Step 1.4.6.5

Simplify.

$\int_{24}^{39} 20 p \sqrt{(1 + {(\frac{5 \sqrt{x}}{x})}^{2}) x} d x$

Step 1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 1.5.1

Apply the product rule to $\frac{5 \sqrt{x}}{x}$ .

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{{(5 \sqrt{x})}^{2}}{x^{2}}) x} d x$

Step 1.5.2

Apply the product rule to $5 \sqrt{x}$ .

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{5^{2} {\sqrt{x}}^{2}}{x^{2}}) x} d x$

Step 1.6

Simplify the numerator.

Tap for more steps...

Step 1.6.1

Raise $5$ to the power of $2$ .

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{25 {\sqrt{x}}^{2}}{x^{2}}) x} d x$

Step 1.6.2

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

Tap for more steps...

Step 1.6.2.1

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

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{25 {(x^{\frac{1}{2}})}^{2}}{x^{2}}) x} d x$

Step 1.6.2.2

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

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{25 x^{\frac{1}{2} \cdot 2}}{x^{2}}) x} d x$

Step 1.6.2.3

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

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{25 x^{\frac{2}{2}}}{x^{2}}) x} d x$

Step 1.6.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.6.2.4.1

Cancel the common factor.

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{25 x^{\frac{2}{2}}}{x^{2}}) x} d x$

Step 1.6.2.4.2

Rewrite the expression.

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{25 x^{1}}{x^{2}}) x} d x$

Step 1.6.2.5

Simplify.

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{25 x}{x^{2}}) x} d x$

Step 1.7

Cancel the common factor of $x$ and $x^{2}$ .

Tap for more steps...

Step 1.7.1

Factor $x$ out of $25 x$ .

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{x \cdot 25}{x^{2}}) x} d x$

Step 1.7.2

Cancel the common factors.

Tap for more steps...

Step 1.7.2.1

Factor $x$ out of $x^{2}$ .

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{x \cdot 25}{x \cdot x}) x} d x$

Step 1.7.2.2

Cancel the common factor.

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{x \cdot 25}{x \cdot x}) x} d x$

Step 1.7.2.3

Rewrite the expression.

$\int_{24}^{39} 20 p \sqrt{(1 + \frac{25}{x}) x} d x$

Step 1.8

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

$\int_{24}^{39} 20 p \sqrt{(\frac{x}{x} + \frac{25}{x}) x} d x$

Step 1.9

Combine the numerators over the common denominator.

$\int_{24}^{39} 20 p \sqrt{\frac{x + 25}{x} x} d x$

Step 1.10

Combine $\frac{x + 25}{x}$ and $x$ .

$\int_{24}^{39} 20 p \sqrt{\frac{(x + 25) x}{x}} d x$

Step 1.11

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.11.1

Cancel the common factor.

$\int_{24}^{39} 20 p \sqrt{\frac{(x + 25) x}{x}} d x$

Step 1.11.2

Divide $x + 25$ by $1$ .

$\int_{24}^{39} 20 p \sqrt{x + 25} d x$

Step 2

Since $20 p$ is constant with respect to $x$ , move $20 p$ out of the integral.

$20 p \int_{24}^{39} \sqrt{x + 25} d x$

Step 3

Let $u = x + 25$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 3.1

Let $u = x + 25$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 3.1.1

Differentiate $x + 25$ .

$\frac{d}{d x} [x + 25]$

Step 3.1.2

By the Sum Rule, the derivative of $x + 25$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [25]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [25]$

Step 3.1.3

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

$1 + \frac{d}{d x} [25]$

Step 3.1.4

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$1 + 0$

Step 3.1.5

Add $1$ and $0$ .

$1$

Step 3.2

Substitute the lower limit in for $x$ in $u = x + 25$ .

$u_{lower} = 24 + 25$

Step 3.3

Add $24$ and $25$ .

$u_{lower} = 49$

Step 3.4

Substitute the upper limit in for $x$ in $u = x + 25$ .

$u_{upper} = 39 + 25$

Step 3.5

Add $39$ and $25$ .

$u_{upper} = 64$

Step 3.6

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

$u_{lower} = 49$

$u_{upper} = 64$

Step 3.7

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

$20 p \int_{49}^{64} \sqrt{u} d u$

Step 4

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

$20 p \int_{49}^{64} u^{\frac{1}{2}} d u$

Step 5

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

$20 p \frac{2}{3} u^{\frac{3}{2}}]_{49}^{64}$

Step 6

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

$20 p \frac{2 u^{\frac{3}{2}}}{3}]_{49}^{64}$

Step 7

Substitute and simplify.

Tap for more steps...

Step 7.1

Evaluate $\frac{2 u^{\frac{3}{2}}}{3}$ at $64$ and at $49$ .

$20 p ((\frac{2 \cdot 64^{\frac{3}{2}}}{3}) - \frac{2 \cdot 49^{\frac{3}{2}}}{3})$

Step 7.2

Simplify.

Tap for more steps...

Step 7.2.1

Rewrite $64$ as $8^{2}$ .

$20 p (\frac{2 \cdot {(8^{2})}^{\frac{3}{2}}}{3} - \frac{2 \cdot 49^{\frac{3}{2}}}{3})$

Step 7.2.2

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

$20 p (\frac{2 \cdot 8^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 49^{\frac{3}{2}}}{3})$

Step 7.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.2.3.1

Cancel the common factor.

$20 p (\frac{2 \cdot 8^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 49^{\frac{3}{2}}}{3})$

Step 7.2.3.2

Rewrite the expression.

$20 p (\frac{2 \cdot 8^{3}}{3} - \frac{2 \cdot 49^{\frac{3}{2}}}{3})$

Step 7.2.4

Raise $8$ to the power of $3$ .

$20 p (\frac{2 \cdot 512}{3} - \frac{2 \cdot 49^{\frac{3}{2}}}{3})$

Step 7.2.5

Multiply $2$ by $512$ .

$20 p (\frac{1024}{3} - \frac{2 \cdot 49^{\frac{3}{2}}}{3})$

Step 7.2.6

Rewrite $49$ as $7^{2}$ .

$20 p (\frac{1024}{3} - \frac{2 \cdot {(7^{2})}^{\frac{3}{2}}}{3})$

Step 7.2.7

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

$20 p (\frac{1024}{3} - \frac{2 \cdot 7^{2 (\frac{3}{2})}}{3})$

Step 7.2.8

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.2.8.1

Cancel the common factor.

$20 p (\frac{1024}{3} - \frac{2 \cdot 7^{2 (\frac{3}{2})}}{3})$

Step 7.2.8.2

Rewrite the expression.

$20 p (\frac{1024}{3} - \frac{2 \cdot 7^{3}}{3})$

Step 7.2.9

Raise $7$ to the power of $3$ .

$20 p (\frac{1024}{3} - \frac{2 \cdot 343}{3})$

Step 7.2.10

Multiply $2$ by $343$ .

$20 p (\frac{1024}{3} - \frac{686}{3})$

Step 7.2.11

Combine the numerators over the common denominator.

$20 p \frac{1024 - 686}{3}$

Step 7.2.12

Subtract $686$ from $1024$ .

$20 p \frac{338}{3}$

Step 7.2.13

Combine $20$ and $\frac{338}{3}$ .

$p \frac{20 \cdot 338}{3}$

Step 7.2.14

Multiply $20$ by $338$ .

$p \frac{6760}{3}$

Step 7.2.15

Combine $p$ and $\frac{6760}{3}$ .

$\frac{p \cdot 6760}{3}$

Step 7.2.16

Move $6760$ to the left of $p$ .

$\frac{6760 p}{3}$

Step 8

Reorder terms.

$\frac{6760}{3} p$

Step 9

Combine $\frac{6760}{3}$ and $p$ .

$\frac{6760 p}{3}$

Step 10