Calculus Examples

$\frac{1}{5 \int_{0}^{5} 5.3 \sin^{2} (w t) d t}$

Step 1

Since $5.3$ is constant with respect to $t$ , move $5.3$ out of the integral.

$\frac{1}{5 (5.3 \int_{0}^{5} \sin^{2} (w t) d t)}$

Step 2

Multiply $5.3$ by $5$ .

$\frac{1}{26.5 \int_{0}^{5} \sin^{2} (w t) d t}$

Step 3

Let $u_{1} = w t$ . Then $d u_{1} = w d t$ , so $\frac{1}{w} d u_{1} = d t$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 3.1

Let $u_{1} = w t$ . Find $\frac{d u_{1}}{d t}$ .

Tap for more steps...

Step 3.1.1

Differentiate $w t$ .

$\frac{d}{d t} [w t]$

Step 3.1.2

Since $w$ is constant with respect to $t$ , the derivative of $w t$ with respect to $t$ is $w \frac{d}{d t} [t]$ .

$w \frac{d}{d t} [t]$

Step 3.1.3

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

$w \cdot 1$

Step 3.1.4

Multiply $w$ by $1$ .

$w$

Step 3.2

Substitute the lower limit in for $t$ in $u_{1} = w t$ .

$u_{lower} = w \cdot 0$

Step 3.3

Multiply $w$ by $0$ .

$u_{lower} = 0$

Step 3.4

Substitute the upper limit in for $t$ in $u_{1} = w t$ .

$u_{upper} = w \cdot 5$

Step 3.5

Move $5$ to the left of $w$ .

$u_{upper} = 5 w$

Step 3.6

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

$u_{lower} = 0$

$u_{upper} = 5 w$

Step 3.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$\frac{1}{26.5 \int_{0}^{5 w} \sin^{2} (u_{1}) \frac{1}{w} d u_{1}}$

Step 4

Combine $\sin^{2} (u_{1})$ and $\frac{1}{w}$ .

$\frac{1}{26.5 \int_{0}^{5 w} \frac{\sin^{2} (u_{1})}{w} d u_{1}}$

Step 5

Since $\frac{1}{w}$ is constant with respect to $u_{1}$ , move $\frac{1}{w}$ out of the integral.

$\frac{1}{26.5 (\frac{1}{w} \int_{0}^{5 w} \sin^{2} (u_{1}) d u_{1})}$

Step 6

Combine $\frac{1}{w}$ and $26.5$ .

$\frac{1}{\frac{26.5}{w} \int_{0}^{5 w} \sin^{2} (u_{1}) d u_{1}}$

Step 7

Use the half-angle formula to rewrite $\sin^{2} (u_{1})$ as $\frac{1 - \cos (2 u_{1})}{2}$ .

$\frac{1}{\frac{26.5}{w} \int_{0}^{5 w} \frac{1 - \cos (2 u_{1})}{2} d u_{1}}$

Step 8

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{\frac{26.5}{w} (\frac{1}{2} \int_{0}^{5 w} 1 - \cos (2 u_{1}) d u_{1})}$

Step 9

Multiply $\frac{1}{2}$ by $\frac{26.5}{w}$ .

$\frac{1}{\frac{26.5}{2 w} \int_{0}^{5 w} 1 - \cos (2 u_{1}) d u_{1}}$

Step 10

Split the single integral into multiple integrals.

$\frac{1}{\frac{26.5}{2 w} (\int_{0}^{5 w} d u_{1} + \int_{0}^{5 w} - \cos (2 u_{1}) d u_{1})}$

Step 11

Apply the constant rule.

$\frac{1}{\frac{26.5}{2 w} (u_{1}]_{0}^{5 w} + \int_{0}^{5 w} - \cos (2 u_{1}) d u_{1})}$

Step 12

Since $- 1$ is constant with respect to $u_{1}$ , move $- 1$ out of the integral.

$\frac{1}{\frac{26.5}{2 w} (u_{1}]_{0}^{5 w} - \int_{0}^{5 w} \cos (2 u_{1}) d u_{1})}$

Step 13

Let $u_{2} = 2 u_{1}$ . Then $d u_{2} = 2 d u_{1}$ , so $\frac{1}{2} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 13.1

Let $u_{2} = 2 u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

Tap for more steps...

Step 13.1.1

Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Step 13.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

Step 13.1.3

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

$2 \cdot 1$

Step 13.1.4

Multiply $2$ by $1$ .

$2$

Step 13.2

Substitute the lower limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{lower} = 2 \cdot 0$

Step 13.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 13.4

Substitute the upper limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{upper} = 2 (5 w)$

Step 13.5

Multiply $5$ by $2$ .

$u_{upper} = 10 w$

Step 13.6

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

$u_{lower} = 0$

$u_{upper} = 10 w$

Step 13.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\frac{1}{\frac{26.5}{2 w} (u_{1}]_{0}^{5 w} - \int_{0}^{10 w} \cos (u_{2}) \frac{1}{2} d u_{2})}$

Step 14

Combine $\cos (u_{2})$ and $\frac{1}{2}$ .

$\frac{1}{\frac{26.5}{2 w} (u_{1}]_{0}^{5 w} - \int_{0}^{10 w} \frac{\cos (u_{2})}{2} d u_{2})}$

Step 15

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{\frac{26.5}{2 w} (u_{1}]_{0}^{5 w} - (\frac{1}{2} \int_{0}^{10 w} \cos (u_{2}) d u_{2}))}$

Step 16

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{\frac{26.5}{2 w} (u_{1}]_{0}^{5 w} - \frac{1}{2} \sin (u_{2})]_{0}^{10 w})}$

Step 17

Combine $\sin (u_{2})]_{0}^{10 w}$ and $\frac{1}{2}$ .

$\frac{1}{\frac{26.5}{2 w} (u_{1}]_{0}^{5 w} - \frac{\sin (u_{2})]_{0}^{10 w}}{2})}$

Step 18

Substitute and simplify.

Tap for more steps...

Step 18.1

Evaluate $u_{1}$ at $5 w$ and at $0$ .

$\frac{1}{\frac{26.5}{2 w} ((5 w) + 0 - \frac{\sin (u_{2})]_{0}^{10 w}}{2})}$

Step 18.2

Evaluate $\sin (u_{2})$ at $10 w$ and at $0$ .

$\frac{1}{\frac{26.5}{2 w} ((5 w) + 0 - \frac{(\sin (10 w)) - \sin (0)}{2})}$

Step 18.3

Add $5 w$ and $0$ .

$\frac{1}{\frac{26.5}{2 w} (5 w - \frac{\sin (10 w) - \sin (0)}{2})}$

Step 19

Simplify.

Tap for more steps...

Step 19.1

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{\frac{26.5}{2 w} (5 w - \frac{\sin (10 w) - 0}{2})}$

Step 19.2

Multiply $- 1$ by $0$ .

$\frac{1}{\frac{26.5}{2 w} (5 w - \frac{\sin (10 w) + 0}{2})}$

Step 19.3

Add $\sin (10 w)$ and $0$ .

$\frac{1}{\frac{26.5}{2 w} (5 w - \frac{\sin (10 w)}{2})}$

Step 20

Simplify.

Tap for more steps...

Step 20.1

Factor $26.5$ out of $26.5$ .

$\frac{1}{\frac{26.5 (1)}{2 w} (5 w - \frac{\sin (10 w)}{2})}$

Step 20.2

Factor $2$ out of $2 w$ .

$\frac{1}{\frac{26.5 (1)}{2 (w)} (5 w - \frac{\sin (10 w)}{2})}$

Step 20.3

Separate fractions.

$\frac{1}{\frac{26.5}{2} \cdot \frac{1}{w} (5 w - \frac{\sin (10 w)}{2})}$

Step 20.4

Divide $26.5$ by $2$ .

$\frac{1}{13.25 \frac{1}{w} (5 w - \frac{\sin (10 w)}{2})}$

Step 20.5

Combine $13.25$ and $\frac{1}{w}$ .

$\frac{1}{\frac{13.25}{w} (5 w - \frac{\sin (10 w)}{2})}$

Step 20.6

Factor $\frac{13.25}{w}$ out of $\frac{1}{\frac{13.25}{w} (5 w - \frac{\sin (10 w)}{2})}$ .

$\frac{w}{13.25} \cdot \frac{1}{5 w - \frac{\sin (10 w)}{2}}$

Step 20.7

Multiply by $1$ .

$\frac{1 w}{13.25} \cdot \frac{1}{5 w - \frac{\sin (10 w)}{2}}$

Step 20.8

Factor $13.25$ out of $13.25$ .

$\frac{1 w}{13.25 (1)} \cdot \frac{1}{5 w - \frac{\sin (10 w)}{2}}$

Step 20.9

Separate fractions.

$\frac{1}{13.25} \cdot \frac{w}{1} \frac{1}{5 w - \frac{\sin (10 w)}{2}}$

Step 20.10

Divide $1$ by $13.25$ .

$0.07547169 \frac{w}{1} \frac{1}{5 w - \frac{\sin (10 w)}{2}}$

Step 20.11

Divide $w$ by $1$ .

$0.07547169 w \frac{1}{5 w - \frac{\sin (10 w)}{2}}$

Step 20.12

Multiply $0.07547169 w \frac{1}{5 w - \frac{\sin (10 w)}{2}}$ .

Tap for more steps...

Step 20.12.1

Combine $0.07547169$ and $\frac{1}{5 w - \frac{\sin (10 w)}{2}}$ .

$w \frac{0.07547169}{5 w - \frac{\sin (10 w)}{2}}$

Step 20.12.2

Combine $w$ and $\frac{0.07547169}{5 w - \frac{\sin (10 w)}{2}}$ .

$\frac{w \cdot 0.07547169}{5 w - \frac{\sin (10 w)}{2}}$

Step 20.13

Move $0.07547169$ to the left of $w$ .

$\frac{0.07547169 w}{5 w - \frac{\sin (10 w)}{2}}$

Step 20.14

Reorder terms.

$\frac{0.07547169 w}{5 w - \frac{1}{2} \sin (10 w)}$

Step 21

Combine $\sin (10 w)$ and $\frac{1}{2}$ .

$\frac{0.07547169 w}{5 w - \frac{\sin (10 w)}{2}}$