Calculus Examples

$\int_{- 3}^{8} (3 \sin (2 x)) d x$

Step 1

Remove parentheses.

$\int_{- 3}^{8} 3 \sin (2 x) d x$

Step 2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{- 3}^{8} \sin (2 x) d x$

Step 3

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

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Step 3.1

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

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Step 3.1.1

Differentiate $2 x$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

$2 \cdot 1$

Step 3.1.4

Multiply $2$ by $1$ .

$2$

Step 3.2

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

$u_{lower} = 2 \cdot - 3$

Step 3.3

Multiply $2$ by $- 3$ .

$u_{lower} = - 6$

Step 3.4

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

$u_{upper} = 2 \cdot 8$

Step 3.5

Multiply $2$ by $8$ .

$u_{upper} = 16$

Step 3.6

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

$u_{lower} = - 6$

$u_{upper} = 16$

Step 3.7

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

$3 \int_{- 6}^{16} \sin (u) \frac{1}{2} d u$

Step 4

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

$3 \int_{- 6}^{16} \frac{\sin (u)}{2} d u$

Step 5

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

$3 (\frac{1}{2} \int_{- 6}^{16} \sin (u) d u)$

Step 6

Combine $\frac{1}{2}$ and $3$ .

$\frac{3}{2} \int_{- 6}^{16} \sin (u) d u$

Step 7

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

$\frac{3}{2} - \cos (u)]_{- 6}^{16}$

Step 8

Evaluate $- \cos (u)$ at $16$ and at $- 6$ .

$\frac{3}{2} (- \cos (16) + \cos (- 6))$

Step 9

Simplify.

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Step 9.1

Simplify each term.

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Step 9.1.1

Evaluate $\cos (16)$ .

$\frac{3}{2} (- - 0.95765948 + \cos (- 6))$

Step 9.1.2

Multiply $- 1$ by $- 0.95765948$ .

$\frac{3}{2} (0.95765948 + \cos (- 6))$

Step 9.1.3

Evaluate $\cos (- 6)$ .

$\frac{3}{2} (0.95765948 + 0.96017028)$

Step 9.2

Add $0.95765948$ and $0.96017028$ .

$\frac{3}{2} \cdot 1.91782976$

Step 9.3

Multiply $\frac{3}{2} \cdot 1.91782976$ .

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Step 9.3.1

Combine $\frac{3}{2}$ and $1.91782976$ .

$\frac{3 \cdot 1.91782976}{2}$

Step 9.3.2

Multiply $3$ by $1.91782976$ .

$\frac{5.7534893}{2}$

Step 9.4

Divide $5.7534893$ by $2$ .

$2.87674465$