Calculus Examples

$\int_{0}^{5 π} - 3 \sin (x) + 4 \cos (x) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{5 π} - 3 \sin (x) d x + \int_{0}^{5 π} 4 \cos (x) d x$

Step 2

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

$- 3 \int_{0}^{5 π} \sin (x) d x + \int_{0}^{5 π} 4 \cos (x) d x$

Step 3

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

$- 3 (- \cos (x)]_{0}^{5 π}) + \int_{0}^{5 π} 4 \cos (x) d x$

Step 4

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$- 3 (- \cos (x)]_{0}^{5 π}) + 4 \int_{0}^{5 π} \cos (x) d x$

Step 5

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

$- 3 (- \cos (x)]_{0}^{5 π}) + 4 (\sin (x)]_{0}^{5 π})$

Step 6

Simplify the answer.

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Step 6.1

Substitute and simplify.

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Step 6.1.1

Evaluate $- \cos (x)$ at $5 π$ and at $0$ .

$- 3 ((- \cos (5 π)) + \cos (0)) + 4 (\sin (x)]_{0}^{5 π})$

Step 6.1.2

Evaluate $\sin (x)$ at $5 π$ and at $0$ .

$- 3 (- \cos (5 π) + \cos (0)) + 4 (\sin (5 π) - \sin (0))$

Step 6.2

Simplify.

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Step 6.2.1

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

$- 3 (- \cos (5 π) + 1) + 4 (\sin (5 π) - \sin (0))$

Step 6.2.2

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

$- 3 (- \cos (5 π) + 1) + 4 (\sin (5 π) - 0)$

Step 6.2.3

Multiply $- 1$ by $0$ .

$- 3 (- \cos (5 π) + 1) + 4 (\sin (5 π) + 0)$

Step 6.2.4

Add $\sin (5 π)$ and $0$ .

$- 3 (- \cos (5 π) + 1) + 4 \sin (5 π)$

Step 6.3

Simplify.

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Step 6.3.1

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

$- 3 (- \cos (π) + 1) + 4 \sin (5 π)$

Step 6.3.2

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.

$- 3 (- - \cos (0) + 1) + 4 \sin (5 π)$

Step 6.3.3

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

$- 3 (- (- 1 \cdot 1) + 1) + 4 \sin (5 π)$

Step 6.3.4

Multiply $- 1$ by $1$ .

$- 3 (- - 1 + 1) + 4 \sin (5 π)$

Step 6.3.5

Multiply $- 1$ by $- 1$ .

$- 3 (1 + 1) + 4 \sin (5 π)$

Step 6.3.6

Add $1$ and $1$ .

$- 3 \cdot 2 + 4 \sin (5 π)$

Step 6.3.7

Multiply $- 3$ by $2$ .

$- 6 + 4 \sin (5 π)$

Step 6.3.8

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

$- 6 + 4 \sin (π)$

Step 6.3.9

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- 6 + 4 \sin (0)$

Step 6.3.10

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

$- 6 + 4 \cdot 0$

Step 6.3.11

Multiply $4$ by $0$ .

$- 6 + 0$

Step 6.3.12

Add $- 6$ and $0$ .

$- 6$