Calculus Examples

$\int_{0}^{π} 7 \sin (x) + \sin (7 x) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{π} 7 \sin (x) d x + \int_{0}^{π} \sin (7 x) d x$

Step 2

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int_{0}^{π} \sin (x) d x + \int_{0}^{π} \sin (7 x) d x$

Step 3

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

$7 (- \cos (x)]_{0}^{π}) + \int_{0}^{π} \sin (7 x) d x$

Step 4

Let $u = 7 x$ . Then $d u = 7 d x$ , so $\frac{1}{7} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 7 x$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $7 x$ .

$\frac{d}{d x} [7 x]$

Step 4.1.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$7 \frac{d}{d x} [x]$

Step 4.1.3

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

$7 \cdot 1$

Step 4.1.4

Multiply $7$ by $1$ .

$7$

Step 4.2

Substitute the lower limit in for $x$ in $u = 7 x$ .

$u_{lower} = 7 \cdot 0$

Step 4.3

Multiply $7$ by $0$ .

$u_{lower} = 0$

Step 4.4

Substitute the upper limit in for $x$ in $u = 7 x$ .

$u_{upper} = 7 π$

Step 4.5

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

$u_{lower} = 0$

$u_{upper} = 7 π$

Step 4.6

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

$7 (- \cos (x)]_{0}^{π}) + \int_{0}^{7 π} \sin (u) \frac{1}{7} d u$

Step 5

Combine $\sin (u)$ and $\frac{1}{7}$ .

$7 (- \cos (x)]_{0}^{π}) + \int_{0}^{7 π} \frac{\sin (u)}{7} d u$

Step 6

Since $\frac{1}{7}$ is constant with respect to $u$ , move $\frac{1}{7}$ out of the integral.

$7 (- \cos (x)]_{0}^{π}) + \frac{1}{7} \int_{0}^{7 π} \sin (u) d u$

Step 7

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

$7 (- \cos (x)]_{0}^{π}) + \frac{1}{7} - \cos (u)]_{0}^{7 π}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $- \cos (x)$ at $π$ and at $0$ .

$7 ((- \cos (π)) + \cos (0)) + \frac{1}{7} - \cos (u)]_{0}^{7 π}$

Step 8.2

Evaluate $- \cos (u)$ at $7 π$ and at $0$ .

$7 ((- \cos (π)) + \cos (0)) + \frac{1}{7} ((- \cos (7 π)) + \cos (0))$

Step 8.3

Remove parentheses.

$7 (- \cos (π) + \cos (0)) + \frac{1}{7} (- \cos (7 π) + \cos (0))$

Step 9

Simplify.

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Step 9.1

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

$7 (- \cos (π) + 1) + \frac{1}{7} (- \cos (7 π) + \cos (0))$

Step 9.2

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

$7 (- \cos (π) + 1) + \frac{1}{7} (- \cos (7 π) + 1)$

Step 10

Simplify.

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Step 10.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$7 (- - \cos (0) + 1) + \frac{1}{7} (- \cos (7 π) + 1)$

Step 10.2

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

$7 (- (- 1 \cdot 1) + 1) + \frac{1}{7} (- \cos (7 π) + 1)$

Step 10.3

Multiply $- 1$ by $1$ .

$7 (- - 1 + 1) + \frac{1}{7} (- \cos (7 π) + 1)$

Step 10.4

Multiply $- 1$ by $- 1$ .

$7 (1 + 1) + \frac{1}{7} (- \cos (7 π) + 1)$

Step 10.5

Add $1$ and $1$ .

$7 \cdot 2 + \frac{1}{7} (- \cos (7 π) + 1)$

Step 10.6

Multiply $7$ by $2$ .

$14 + \frac{1}{7} (- \cos (7 π) + 1)$

Step 10.7

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$14 + \frac{1}{7} (- \cos (π) + 1)$

Step 10.8

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$14 + \frac{1}{7} (- - \cos (0) + 1)$

Step 10.9

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

$14 + \frac{1}{7} (- (- 1 \cdot 1) + 1)$

Step 10.10

Multiply $- 1$ by $1$ .

$14 + \frac{1}{7} (- - 1 + 1)$

Step 10.11

Multiply $- 1$ by $- 1$ .

$14 + \frac{1}{7} (1 + 1)$

Step 10.12

Add $1$ and $1$ .

$14 + \frac{1}{7} \cdot 2$

Step 10.13

Combine $\frac{1}{7}$ and $2$ .

$14 + \frac{2}{7}$

Step 10.14

To write $14$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$14 \cdot \frac{7}{7} + \frac{2}{7}$

Step 10.15

Combine $14$ and $\frac{7}{7}$ .

$\frac{14 \cdot 7}{7} + \frac{2}{7}$

Step 10.16

Combine the numerators over the common denominator.

$\frac{14 \cdot 7 + 2}{7}$

Step 10.17

Simplify the numerator.

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Step 10.17.1

Multiply $14$ by $7$ .

$\frac{98 + 2}{7}$

Step 10.17.2

Add $98$ and $2$ .

$\frac{100}{7}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{100}{7}$

Decimal Form:

$14.28571428 \dots$

Mixed Number Form:

$14 \frac{2}{7}$