Calculus Examples

$\int_{0}^{\frac{π}{2}} \cos (14 x) d x$

Step 1

Let $u = 14 x$ . Then $d u = 14 d x$ , so $\frac{1}{14} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 14 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $14 x$ .

$\frac{d}{d x} [14 x]$

Step 1.1.2

Since $14$ is constant with respect to $x$ , the derivative of $14 x$ with respect to $x$ is $14 \frac{d}{d x} [x]$ .

$14 \frac{d}{d x} [x]$

Step 1.1.3

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

$14 \cdot 1$

Step 1.1.4

Multiply $14$ by $1$ .

$14$

Step 1.2

Substitute the lower limit in for $x$ in $u = 14 x$ .

$u_{lower} = 14 \cdot 0$

Step 1.3

Multiply $14$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = 14 x$ .

$u_{upper} = 14 \frac{π}{2}$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $14$ .

$u_{upper} = 2 (7) \frac{π}{2}$

Step 1.5.2

Cancel the common factor.

$u_{upper} = 2 \cdot 7 \frac{π}{2}$

Step 1.5.3

Rewrite the expression.

$u_{upper} = 7 π$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 7 π$

Step 1.7

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

$\int_{0}^{7 π} \cos (u) \frac{1}{14} d u$

Step 2

Combine $\cos (u)$ and $\frac{1}{14}$ .

$\int_{0}^{7 π} \frac{\cos (u)}{14} d u$

Step 3

Since $\frac{1}{14}$ is constant with respect to $u$ , move $\frac{1}{14}$ out of the integral.

$\frac{1}{14} \int_{0}^{7 π} \cos (u) d u$

Step 4

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

$\frac{1}{14} \sin (u)]_{0}^{7 π}$

Step 5

Evaluate $\sin (u)$ at $7 π$ and at $0$ .

$\frac{1}{14} (\sin (7 π) - \sin (0))$

Step 6

Simplify.

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Step 6.1

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

$\frac{1}{14} (\sin (7 π) - 0)$

Step 6.2

Multiply $- 1$ by $0$ .

$\frac{1}{14} (\sin (7 π) + 0)$

Step 6.3

Add $\sin (7 π)$ and $0$ .

$\frac{1}{14} \sin (7 π)$

Step 6.4

Combine $\frac{1}{14}$ and $\sin (7 π)$ .

$\frac{\sin (7 π)}{14}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{\sin (π)}{14}$

Step 7.1.2

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

$\frac{\sin (0)}{14}$

Step 7.1.3

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

$\frac{0}{14}$

Step 7.2

Divide $0$ by $14$ .

$0$