Calculus Examples

$\int_{0}^{\frac{π}{4}} 2 \sin (π - x) d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{0}^{\frac{π}{4}} \sin (π - x) d x$

Step 2

Let $u = π - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $π - x$ .

$\frac{d}{d x} [π - x]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $π - x$ with respect to $x$ is $\frac{d}{d x} [π] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [π] + \frac{d}{d x} [- x]$

Step 2.1.2.2

Since $π$ is constant with respect to $x$ , the derivative of $π$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- x]$ .

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Step 2.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$0 - \frac{d}{d x} [x]$

Step 2.1.3.2

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

$0 - 1 \cdot 1$

Step 2.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 2.1.4

Subtract $1$ from $0$ .

$- 1$

Step 2.2

Substitute the lower limit in for $x$ in $u = π - x$ .

$u_{lower} = π - 0$

Step 2.3

Subtract $0$ from $π$ .

$u_{lower} = π$

Step 2.4

Substitute the upper limit in for $x$ in $u = π - x$ .

$u_{upper} = π - \frac{π}{4}$

Step 2.5

Simplify.

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Step 2.5.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$u_{upper} = π \cdot \frac{4}{4} - \frac{π}{4}$

Step 2.5.2

Combine $π$ and $\frac{4}{4}$ .

$u_{upper} = \frac{π \cdot 4}{4} - \frac{π}{4}$

Step 2.5.3

Combine the numerators over the common denominator.

$u_{upper} = \frac{π \cdot 4 - π}{4}$

Step 2.5.4

Simplify the numerator.

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Step 2.5.4.1

Move $4$ to the left of $π$ .

$u_{upper} = \frac{4 \cdot π - π}{4}$

Step 2.5.4.2

Subtract $π$ from $4 π$ .

$u_{upper} = \frac{3 π}{4}$

Step 2.6

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

$u_{lower} = π$

$u_{upper} = \frac{3 π}{4}$

Step 2.7

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

$2 \int_{π}^{\frac{3 π}{4}} - \sin (u) d u$

Step 3

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

$2 (- \int_{π}^{\frac{3 π}{4}} \sin (u) d u)$

Step 4

Multiply $- 1$ by $2$ .

$- 2 \int_{π}^{\frac{3 π}{4}} \sin (u) d u$

Step 5

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

$- 2 (- \cos (u)]_{π}^{\frac{3 π}{4}})$

Step 6

Evaluate $- \cos (u)$ at $\frac{3 π}{4}$ and at $π$ .

$- 2 (- \cos (\frac{3 π}{4}) + \cos (π))$

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

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.

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

Step 7.1.2

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

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

Step 7.1.3

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

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Step 7.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 7.1.3.2

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

$- 2 (\frac{\sqrt{2}}{2} + \cos (π))$

Step 7.1.4

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.

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

Step 7.1.5

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

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

Step 7.1.6

Multiply $- 1$ by $1$ .

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Factor $2$ out of $- 2$ .

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

Step 7.3.2

Cancel the common factor.

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

Step 7.3.3

Rewrite the expression.

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

Step 7.4

Multiply $- 2$ by $- 1$ .

$- \sqrt{2} + 2$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{2} + 2$

Decimal Form:

$0.58578643 \dots$