Calculus Examples

$\int_{- \frac{π}{4}}^{7 \frac{π}{4}} (\sin (x) + \cos (x)) d x$

Step 1

Remove parentheses.

$\int_{- \frac{π}{4}}^{7 (\frac{π}{4})} \sin (x) + \cos (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- \frac{π}{4}}^{7 (\frac{π}{4})} \sin (x) d x + \int_{- \frac{π}{4}}^{7 (\frac{π}{4})} \cos (x) d x$

Step 3

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

$- \cos (x)]_{- \frac{π}{4}}^{7 (\frac{π}{4})} + \int_{- \frac{π}{4}}^{7 (\frac{π}{4})} \cos (x) d x$

Step 4

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

$- \cos (x)]_{- \frac{π}{4}}^{7 (\frac{π}{4})} + \sin (x)]_{- \frac{π}{4}}^{7 (\frac{π}{4})}$

Step 5

Simplify the answer.

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Step 5.1

Combine $- \cos (x)]_{- \frac{π}{4}}^{7 (\frac{π}{4})}$ and $\sin (x)]_{- \frac{π}{4}}^{7 (\frac{π}{4})}$ .

$- \cos (x) + \sin (x)]_{- \frac{π}{4}}^{7 (\frac{π}{4})}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $- \cos (x) + \sin (x)$ at $7 (\frac{π}{4})$ and at $- \frac{π}{4}$ .

$(- \cos (7 \frac{π}{4}) + \sin (7 \frac{π}{4})) - (- \cos (- \frac{π}{4}) + \sin (- \frac{π}{4}))$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Combine $7$ and $\frac{π}{4}$ .

$- \cos (\frac{7 π}{4}) + \sin (7 \frac{π}{4}) - (- \cos (- \frac{π}{4}) + \sin (- \frac{π}{4}))$

Step 5.2.2.2

Combine $7$ and $\frac{π}{4}$ .

$- \cos (\frac{7 π}{4}) + \sin (\frac{7 π}{4}) - (- \cos (- \frac{π}{4}) + \sin (- \frac{π}{4}))$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$- \cos (\frac{π}{4}) + \sin (\frac{7 π}{4}) - (- \cos (- \frac{π}{4}) + \sin (- \frac{π}{4}))$

Step 6.1.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2} + \sin (\frac{7 π}{4}) - (- \cos (- \frac{π}{4}) + \sin (- \frac{π}{4}))$

Step 6.1.3

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

$- \frac{\sqrt{2}}{2} - \sin (\frac{π}{4}) - (- \cos (- \frac{π}{4}) + \sin (- \frac{π}{4}))$

Step 6.1.4

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - (- \cos (- \frac{π}{4}) + \sin (- \frac{π}{4}))$

Step 6.1.5

Simplify each term.

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Step 6.1.5.1

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

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - (- \cos (\frac{7 π}{4}) + \sin (- \frac{π}{4}))$

Step 6.1.5.2

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

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - (- \cos (\frac{π}{4}) + \sin (- \frac{π}{4}))$

Step 6.1.5.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - (- \frac{\sqrt{2}}{2} + \sin (- \frac{π}{4}))$

Step 6.1.5.4

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

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - (- \frac{\sqrt{2}}{2} + \sin (\frac{7 π}{4}))$

Step 6.1.5.5

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

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - (- \frac{\sqrt{2}}{2} - \sin (\frac{π}{4}))$

Step 6.1.5.6

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - (- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2})$

Step 6.1.6

Combine the numerators over the common denominator.

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - \frac{- \sqrt{2} - \sqrt{2}}{2}$

Step 6.1.7

Subtract $\sqrt{2}$ from $- \sqrt{2}$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - \frac{- 2 \sqrt{2}}{2}$

Step 6.1.8

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

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Step 6.1.8.1

Factor $2$ out of $- 2 \sqrt{2}$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - \frac{2 (- \sqrt{2})}{2}$

Step 6.1.8.2

Cancel the common factors.

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Step 6.1.8.2.1

Factor $2$ out of $2$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - \frac{2 (- \sqrt{2})}{2 (1)}$

Step 6.1.8.2.2

Cancel the common factor.

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - \frac{2 (- \sqrt{2})}{2 \cdot 1}$

Step 6.1.8.2.3

Rewrite the expression.

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - \frac{- \sqrt{2}}{1}$

Step 6.1.8.2.4

Divide $- \sqrt{2}$ by $1$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - - \sqrt{2}$

Step 6.1.9

Multiply $- - \sqrt{2}$ .

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Step 6.1.9.1

Multiply $- 1$ by $- 1$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + 1 \sqrt{2}$

Step 6.1.9.2

Multiply $\sqrt{2}$ by $1$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + \sqrt{2}$

Step 6.2

Combine the numerators over the common denominator.

$\sqrt{2} + \frac{- \sqrt{2} - \sqrt{2}}{2}$

Step 6.3

Subtract $\sqrt{2}$ from $- \sqrt{2}$ .

$\sqrt{2} + \frac{- 2 \sqrt{2}}{2}$

Step 6.4

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

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Step 6.4.1

Factor $2$ out of $- 2 \sqrt{2}$ .

$\sqrt{2} + \frac{2 (- \sqrt{2})}{2}$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $2$ out of $2$ .

$\sqrt{2} + \frac{2 (- \sqrt{2})}{2 (1)}$

Step 6.4.2.2

Cancel the common factor.

$\sqrt{2} + \frac{2 (- \sqrt{2})}{2 \cdot 1}$

Step 6.4.2.3

Rewrite the expression.

$\sqrt{2} + \frac{- \sqrt{2}}{1}$

Step 6.4.2.4

Divide $- \sqrt{2}$ by $1$ .

$\sqrt{2} - \sqrt{2}$

Step 6.5

Subtract $\sqrt{2}$ from $\sqrt{2}$ .

$0$