Calculus Examples

$\int_{0}^{1} \sin (3 π t) d t$

Step 1

Let $u = 3 π t$ . Then $d u = 3 π d t$ , so $\frac{1}{3 π} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 π t$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $3 π t$ .

$\frac{d}{d t} [3 π t]$

Step 1.1.2

Since $3 π$ is constant with respect to $t$ , the derivative of $3 π t$ with respect to $t$ is $3 π \frac{d}{d t} [t]$ .

$3 π \frac{d}{d t} [t]$

Step 1.1.3

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

$3 π \cdot 1$

Step 1.1.4

Multiply $3$ by $1$ .

$3 π$

Step 1.2

Substitute the lower limit in for $t$ in $u = 3 π t$ .

$u_{lower} = 3 π \cdot 0$

Step 1.3

Multiply $3 π \cdot 0$ .

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Step 1.3.1

Multiply $0$ by $3$ .

$u_{lower} = 0 π$

Step 1.3.2

Multiply $0$ by $π$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $t$ in $u = 3 π t$ .

$u_{upper} = 3 π \cdot 1$

Step 1.5

Multiply $3$ by $1$ .

$u_{upper} = 3 π$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 3 π$

Step 1.7

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

$\int_{0}^{3 π} \sin (u) \frac{1}{3 π} d u$

Step 2

Combine $\sin (u)$ and $\frac{1}{3 π}$ .

$\int_{0}^{3 π} \frac{\sin (u)}{3 π} d u$

Step 3

Since $\frac{1}{3 π}$ is constant with respect to $u$ , move $\frac{1}{3 π}$ out of the integral.

$\frac{1}{3 π} \int_{0}^{3 π} \sin (u) d u$

Step 4

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

$\frac{1}{3 π} - \cos (u)]_{0}^{3 π}$

Step 5

Evaluate $- \cos (u)$ at $3 π$ and at $0$ .

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

Step 6

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

$\frac{1}{3 π} (- \cos (3 π) + 1)$

Step 7

Simplify.

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Step 7.1

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

$\frac{1}{3 π} (- \cos (π) + 1)$

Step 7.2

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.

$\frac{1}{3 π} (- - \cos (0) + 1)$

Step 7.3

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

$\frac{1}{3 π} (- (- 1 \cdot 1) + 1)$

Step 7.4

Multiply $- 1$ by $1$ .

$\frac{1}{3 π} (- - 1 + 1)$

Step 7.5

Multiply $- 1$ by $- 1$ .

$\frac{1}{3 π} (1 + 1)$

Step 7.6

Add $1$ and $1$ .

$\frac{1}{3 π} \cdot 2$

Step 7.7

Combine $\frac{1}{3 π}$ and $2$ .

$\frac{2}{3 π}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3 π}$

Decimal Form:

$0.21220659 \dots$