Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

$\sin (x)]_{0}^{\frac{π}{4}} + \int_{0}^{\frac{π}{4}} - \sin (x) d x$

Step 3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\sin (x)]_{0}^{\frac{π}{4}} - \int_{0}^{\frac{π}{4}} \sin (x) d x$

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Substitute and simplify.

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Step 5.1.1

Evaluate $\sin (x)$ at $\frac{π}{4}$ and at $0$ .

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

Step 5.1.2

Evaluate $- \cos (x)$ at $\frac{π}{4}$ and at $0$ .

$\sin (\frac{π}{4}) - \sin (0) - (- \cos (\frac{π}{4}) + \cos (0))$

Step 5.2

Simplify.

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Step 5.2.1

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

$\frac{\sqrt{2}}{2} - \sin (0) - (- \cos (\frac{π}{4}) + \cos (0))$

Step 5.2.2

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

$\frac{\sqrt{2}}{2} - 0 - (- \cos (\frac{π}{4}) + \cos (0))$

Step 5.2.3

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

$\frac{\sqrt{2}}{2} - 0 - (- \frac{\sqrt{2}}{2} + \cos (0))$

Step 5.2.4

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

$\frac{\sqrt{2}}{2} - 0 - (- \frac{\sqrt{2}}{2} + 1)$

Step 5.2.5

Multiply $- 1$ by $0$ .

$\frac{\sqrt{2}}{2} + 0 - (- \frac{\sqrt{2}}{2} + 1)$

Step 5.2.6

Add $\frac{\sqrt{2}}{2}$ and $0$ .

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

Step 5.2.7

To write $- (- \frac{\sqrt{2}}{2} + 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.2.8

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

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

Step 5.2.9

Combine the numerators over the common denominator.

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

Step 5.2.10

Multiply $2$ by $- 1$ .

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

Step 5.3

Simplify.

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Cancel the common factor of $2$ .

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Step 5.3.2.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

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

Step 5.3.2.2

Factor $2$ out of $- 2$ .

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

Step 5.3.2.3

Cancel the common factor.

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

Step 5.3.2.4

Rewrite the expression.

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

Step 5.3.3

Multiply $- 1$ by $- 1$ .

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

Step 5.3.4

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

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

Step 5.3.5

Multiply $- 2$ by $1$ .

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

Step 5.3.6

Add $\sqrt{2}$ and $\sqrt{2}$ .

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

Step 5.3.7

Cancel the common factor of $2 \sqrt{2} - 2$ and $2$ .

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Step 5.3.7.1

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

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

Step 5.3.7.2

Factor $2$ out of $- 2$ .

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

Step 5.3.7.3

Factor $2$ out of $2 (\sqrt{2}) + 2 (- 1)$ .

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

Step 5.3.7.4

Cancel the common factors.

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Step 5.3.7.4.1

Factor $2$ out of $2$ .

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

Step 5.3.7.4.2

Cancel the common factor.

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

Step 5.3.7.4.3

Rewrite the expression.

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

Step 5.3.7.4.4

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

$\sqrt{2} - 1$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2} - 1$

Decimal Form:

$0.41421356 \dots$