Calculus Examples

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

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $2 x$ .

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

Step 1.1.2

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 1.1.3

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 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

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

$u_{lower} = 2 (- \frac{π}{6})$

Step 1.3

Simplify.

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Step 1.3.1

Cancel the common factor of $2$ .

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Step 1.3.1.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$u_{lower} = 2 \frac{- π}{6}$

Step 1.3.1.2

Factor $2$ out of $6$ .

$u_{lower} = 2 \frac{- π}{2 (3)}$

Step 1.3.1.3

Cancel the common factor.

$u_{lower} = 2 \frac{- π}{2 \cdot 3}$

Step 1.3.1.4

Rewrite the expression.

$u_{lower} = \frac{- π}{3}$

Step 1.3.2

Move the negative in front of the fraction.

$u_{lower} = - \frac{π}{3}$

Step 1.4

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

$u_{upper} = 2 \frac{π}{6}$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $6$ .

$u_{upper} = 2 \frac{π}{2 (3)}$

Step 1.5.2

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2 \cdot 3}$

Step 1.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{3}$

Step 1.6

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

$u_{lower} = - \frac{π}{3}$

$u_{upper} = \frac{π}{3}$

Step 1.7

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

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

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

$\frac{1}{2} (- \frac{1}{2} + \cos (\frac{5 π}{3}))$

Step 7.1.2

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

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

Step 7.1.3

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

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

Step 7.2

Combine the numerators over the common denominator.

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

Step 7.3

Add $- 1$ and $1$ .

$\frac{1}{2} \cdot \frac{0}{2}$

Step 7.4

Divide $0$ by $2$ .

$\frac{1}{2} \cdot 0$

Step 7.5

Multiply $\frac{1}{2}$ by $0$ .

$0$