Calculus Examples

$\int_{0}^{4.192} 120 \sin (2 t) d t$

Step 1

Since $120$ is constant with respect to $t$ , move $120$ out of the integral.

$120 \int_{0}^{4.192} \sin (2 t) d t$

Step 2

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

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Step 2.1

Let $u = 2 t$ . Find $\frac{d u}{d t}$ .

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Step 2.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 2.1.2

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

$2 \frac{d}{d t} [t]$

Step 2.1.3

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

$2 \cdot 1$

Step 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

Substitute the lower limit in for $t$ in $u = 2 t$ .

$u_{lower} = 2 \cdot 0$

Step 2.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $t$ in $u = 2 t$ .

$u_{upper} = 2 \cdot 4.192$

Step 2.5

Multiply $2$ by $4.192$ .

$u_{upper} = 8.384$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = 8.384$

Step 2.7

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

$120 \int_{0}^{8.384} \sin (u) \frac{1}{2} d u$

Step 3

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

$120 \int_{0}^{8.384} \frac{\sin (u)}{2} d u$

Step 4

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

$120 (\frac{1}{2} \int_{0}^{8.384} \sin (u) d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{2}$ and $120$ .

$\frac{120}{2} \int_{0}^{8.384} \sin (u) d u$

Step 5.2

Cancel the common factor of $120$ and $2$ .

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Step 5.2.1

Factor $2$ out of $120$ .

$\frac{2 \cdot 60}{2} \int_{0}^{8.384} \sin (u) d u$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 60}{2 (1)} \int_{0}^{8.384} \sin (u) d u$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 \cdot 60}{2 \cdot 1} \int_{0}^{8.384} \sin (u) d u$

Step 5.2.2.3

Rewrite the expression.

$\frac{60}{1} \int_{0}^{8.384} \sin (u) d u$

Step 5.2.2.4

Divide $60$ by $1$ .

$60 \int_{0}^{8.384} \sin (u) d u$

Step 6

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$60 (- \cos (u)]_{0}^{8.384})$

Step 7

Evaluate $- \cos (u)$ at $8.384$ and at $0$ .

$60 (- \cos (8.384) + \cos (0))$

Step 8

The exact value of $\cos (0)$ is $1$ .

$60 (- \cos (8.384) + 1)$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$60 (- \cos (8.384)) + 60 \cdot 1$

Step 9.2

Multiply $- 1$ by $60$ .

$- 60 \cos (8.384) + 60 \cdot 1$

Step 9.3

Multiply $60$ by $1$ .

$- 60 \cos (8.384) + 60$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- 60 \cos (8.384) + 60$

Decimal Form:

$90.33295124 \dots$