Calculus Examples

$\int_{- \frac{π}{2}}^{\frac{π}{6}} \cos (2 x) - \sin (x) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- \frac{π}{2}}^{\frac{π}{6}} \cos (2 x) d x + \int_{- \frac{π}{2}}^{\frac{π}{6}} - \sin (x) d x$

Step 2

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 2.1.3

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

$2 \cdot 1$

Step 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

Substitute the lower limit in for $x$ in $u = 2 x$ .

$u_{lower} = 2 (- \frac{π}{2})$

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

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

$u_{lower} = 2 \frac{- π}{2}$

Step 2.3.2

Cancel the common factor.

$u_{lower} = 2 \frac{- π}{2}$

Step 2.3.3

Rewrite the expression.

$u_{lower} = - π$

Step 2.4

Substitute the upper limit in for $x$ in $u = 2 x$ .

$u_{upper} = 2 \frac{π}{6}$

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Factor $2$ out of $6$ .

$u_{upper} = 2 \frac{π}{2 (3)}$

Step 2.5.2

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2 \cdot 3}$

Step 2.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{3}$

Step 2.6

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

$u_{lower} = - π$

$u_{upper} = \frac{π}{3}$

Step 2.7

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

$\int_{- π}^{\frac{π}{3}} \cos (u) \frac{1}{2} d u + \int_{- \frac{π}{2}}^{\frac{π}{6}} - \sin (x) d x$

Step 3

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\int_{- π}^{\frac{π}{3}} \frac{\cos (u)}{2} d u + \int_{- \frac{π}{2}}^{\frac{π}{6}} - \sin (x) d x$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{- π}^{\frac{π}{3}} \cos (u) d u + \int_{- \frac{π}{2}}^{\frac{π}{6}} - \sin (x) d x$

Step 5

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

$\frac{1}{2} \sin (u)]_{- π}^{\frac{π}{3}} + \int_{- \frac{π}{2}}^{\frac{π}{6}} - \sin (x) d x$

Step 6

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

$\frac{1}{2} \sin (u)]_{- π}^{\frac{π}{3}} - \int_{- \frac{π}{2}}^{\frac{π}{6}} \sin (x) d x$

Step 7

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

$\frac{1}{2} \sin (u)]_{- π}^{\frac{π}{3}} - (- \cos (x)]_{- \frac{π}{2}}^{\frac{π}{6}})$

Step 8

Substitute and simplify.

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Step 8.1

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

$\frac{1}{2} ((\sin (\frac{π}{3})) - \sin (- π)) - (- \cos (x)]_{- \frac{π}{2}}^{\frac{π}{6}})$

Step 8.2

Evaluate $- \cos (x)$ at $\frac{π}{6}$ and at $- \frac{π}{2}$ .

$\frac{1}{2} ((\sin (\frac{π}{3})) - \sin (- π)) - ((- \cos (\frac{π}{6})) + \cos (- \frac{π}{2}))$

Step 8.3

Remove parentheses.

$\frac{1}{2} (\sin (\frac{π}{3}) - \sin (- π)) - (- \cos (\frac{π}{6}) + \cos (- \frac{π}{2}))$

Step 9

Simplify.

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Step 9.1

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{2} (\frac{\sqrt{3}}{2} - \sin (- π)) - (- \cos (\frac{π}{6}) + \cos (- \frac{π}{2}))$

Step 9.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{2} (\frac{\sqrt{3}}{2} - \sin (- π)) - (- \frac{\sqrt{3}}{2} + \cos (- \frac{π}{2}))$

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.

$\frac{1}{2} (\frac{\sqrt{3}}{2} - \sin (0)) - (- \frac{\sqrt{3}}{2} + \cos (- \frac{π}{2}))$

Step 10.2

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

$\frac{1}{2} (\frac{\sqrt{3}}{2} - 0) - (- \frac{\sqrt{3}}{2} + \cos (- \frac{π}{2}))$

Step 10.3

Multiply $- 1$ by $0$ .

$\frac{1}{2} (\frac{\sqrt{3}}{2} + 0) - (- \frac{\sqrt{3}}{2} + \cos (- \frac{π}{2}))$

Step 10.4

Add $\frac{\sqrt{3}}{2}$ and $0$ .

$\frac{1}{2} \cdot \frac{\sqrt{3}}{2} - (- \frac{\sqrt{3}}{2} + \cos (- \frac{π}{2}))$

Step 10.5

Multiply $\frac{1}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 10.5.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{3}}{2}$ .

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

Step 10.5.2

Multiply $2$ by $2$ .

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

Step 10.6

Simplify each term.

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Step 10.6.1

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

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

Step 10.6.2

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

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

Step 10.6.3

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{\sqrt{3}}{4} - (- \frac{\sqrt{3}}{2} + 0)$

Step 10.7

Add $- \frac{\sqrt{3}}{2}$ and $0$ .

$\frac{\sqrt{3}}{4} - - \frac{\sqrt{3}}{2}$

Step 10.8

Multiply $- - \frac{\sqrt{3}}{2}$ .

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Step 10.8.1

Multiply $- 1$ by $- 1$ .

$\frac{\sqrt{3}}{4} + 1 \frac{\sqrt{3}}{2}$

Step 10.8.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$\frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{2}$

Step 10.9

To write $\frac{\sqrt{3}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{2}{2}$

Step 10.10

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.10.1

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{2}{2}$ .

$\frac{\sqrt{3}}{4} + \frac{\sqrt{3} \cdot 2}{2 \cdot 2}$

Step 10.10.2

Multiply $2$ by $2$ .

$\frac{\sqrt{3}}{4} + \frac{\sqrt{3} \cdot 2}{4}$

Step 10.11

Combine the numerators over the common denominator.

$\frac{\sqrt{3} + \sqrt{3} \cdot 2}{4}$

Step 10.12

Reorder the factors of $\sqrt{3} \cdot 2$ .

$\frac{\sqrt{3} + 2 \sqrt{3}}{4}$

Step 10.13

Add $\sqrt{3}$ and $2 \sqrt{3}$ .

$\frac{3 \sqrt{3}}{4}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{3}}{4}$

Decimal Form:

$1.29903810 \dots$