Calculus Examples

$\int_{0}^{\frac{π}{6}} (1 - \cos (3 t)) \sin (3 t) d t$

Step 1

Let $u_{2} = \cos (3 t)$ . Then $d u_{2} = - 3 \sin (3 t) d t$ , so $- \frac{1}{3} d u_{2} = \sin (3 t) d t$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 1.1

Let $u_{2} = \cos (3 t)$ . Find $\frac{d u_{2}}{d t}$ .

Tap for more steps...

Step 1.1.1

Differentiate $\cos (3 t)$ .

$\frac{d}{d t} [\cos (3 t)]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cos (t)$ and $g (t) = 3 t$ .

Tap for more steps...

Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $3 t$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d t} [3 t]$

Step 1.1.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- \sin (u_{1}) \frac{d}{d t} [3 t]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $3 t$ .

$- \sin (3 t) \frac{d}{d t} [3 t]$

Step 1.1.3

Differentiate.

Tap for more steps...

Step 1.1.3.1

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

$- \sin (3 t) (3 \frac{d}{d t} [t])$

Step 1.1.3.2

Multiply $3$ by $- 1$ .

$- 3 \sin (3 t) \frac{d}{d t} [t]$

Step 1.1.3.3

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

$- 3 \sin (3 t) \cdot 1$

Step 1.1.3.4

Multiply $- 3$ by $1$ .

$- 3 \sin (3 t)$

Step 1.2

Substitute the lower limit in for $t$ in $u_{2} = \cos (3 t)$ .

$u_{lower} = \cos (3 \cdot 0)$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Multiply $3$ by $0$ .

$u_{lower} = \cos (0)$

Step 1.3.2

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

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $t$ in $u_{2} = \cos (3 t)$ .

$u_{upper} = \cos (3 \frac{π}{6})$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.5.1.1

Factor $3$ out of $6$ .

$u_{upper} = \cos (3 \frac{π}{3 (2)})$

Step 1.5.1.2

Cancel the common factor.

$u_{upper} = \cos (3 \frac{π}{3 \cdot 2})$

Step 1.5.1.3

Rewrite the expression.

$u_{upper} = \cos (\frac{π}{2})$

Step 1.5.2

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

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 0$

Step 1.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\int_{1}^{0} (1 - u_{2}) \frac{1}{- 3} d u_{2}$

Step 2

Move the negative in front of the fraction.

$\int_{1}^{0} (1 - u_{2}) (- \frac{1}{3}) d u_{2}$

Step 3

Multiply $(1 - u_{2}) (- \frac{1}{3})$ .

$\int_{1}^{0} 1 (- \frac{1}{3}) - u_{2} (- \frac{1}{3}) d u_{2}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Multiply $- 1$ by $1$ .

$\int_{1}^{0} - 1 (\frac{1}{3}) - u_{2} (- \frac{1}{3}) d u_{2}$

Step 4.2

Rewrite $- 1 (\frac{1}{3})$ as $- (\frac{1}{3})$ .

$\int_{1}^{0} - (\frac{1}{3}) - u_{2} (- \frac{1}{3}) d u_{2}$

Step 4.3

Multiply $- 1$ by $- 1$ .

$\int_{1}^{0} - \frac{1}{3} + 1 u_{2} \frac{1}{3} d u_{2}$

Step 4.4

Multiply $u_{2}$ by $1$ .

$\int_{1}^{0} - \frac{1}{3} + u_{2} \frac{1}{3} d u_{2}$

Step 4.5

Combine $u_{2}$ and $\frac{1}{3}$ .

$\int_{1}^{0} - \frac{1}{3} + \frac{u_{2}}{3} d u_{2}$

Step 5

Split the single integral into multiple integrals.

$\int_{1}^{0} - \frac{1}{3} d u_{2} + \int_{1}^{0} \frac{u_{2}}{3} d u_{2}$

Step 6

Apply the constant rule.

$- \frac{1}{3} u_{2}]_{1}^{0} + \int_{1}^{0} \frac{u_{2}}{3} d u_{2}$

Step 7

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

$- \frac{1}{3} u_{2}]_{1}^{0} + \frac{1}{3} \int_{1}^{0} u_{2} d u_{2}$

Step 8

By the Power Rule, the integral of $u_{2}$ with respect to $u_{2}$ is $\frac{1}{2} {u_{2}}^{2}$ .

$- \frac{1}{3} u_{2}]_{1}^{0} + \frac{1}{3} \frac{1}{2} {u_{2}}^{2}]_{1}^{0}$

Step 9

Substitute and simplify.

Tap for more steps...

Step 9.1

Evaluate $- \frac{1}{3} u_{2}$ at $0$ and at $1$ .

$(- \frac{1}{3} \cdot 0) + \frac{1}{3} \cdot 1 + \frac{1}{3} \frac{1}{2} {u_{2}}^{2}]_{1}^{0}$

Step 9.2

Evaluate $\frac{1}{2} {u_{2}}^{2}$ at $0$ and at $1$ .

$- \frac{1}{3} \cdot 0 + \frac{1}{3} \cdot 1 + \frac{1}{3} (\frac{1}{2} \cdot 0^{2} - \frac{1}{2} \cdot 1^{2})$

Step 9.3

Simplify.

Tap for more steps...

Step 9.3.1

Multiply $0$ by $- 1$ .

$0 (\frac{1}{3}) + \frac{1}{3} \cdot 1 + \frac{1}{3} (\frac{1}{2} \cdot 0^{2} - \frac{1}{2} \cdot 1^{2})$

Step 9.3.2

Multiply $0$ by $\frac{1}{3}$ .

$0 + \frac{1}{3} \cdot 1 + \frac{1}{3} (\frac{1}{2} \cdot 0^{2} - \frac{1}{2} \cdot 1^{2})$

Step 9.3.3

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

$0 + \frac{1}{3} + \frac{1}{3} (\frac{1}{2} \cdot 0^{2} - \frac{1}{2} \cdot 1^{2})$

Step 9.3.4

Add $0$ and $\frac{1}{3}$ .

$\frac{1}{3} + \frac{1}{3} (\frac{1}{2} \cdot 0^{2} - \frac{1}{2} \cdot 1^{2})$

Step 9.3.5

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} + \frac{1}{3} (\frac{1}{2} \cdot 0 - \frac{1}{2} \cdot 1^{2})$

Step 9.3.6

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

$\frac{1}{3} + \frac{1}{3} (0 - \frac{1}{2} \cdot 1^{2})$

Step 9.3.7

One to any power is one.

$\frac{1}{3} + \frac{1}{3} (0 - \frac{1}{2} \cdot 1)$

Step 9.3.8

Multiply $- 1$ by $1$ .

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

Step 9.3.9

Subtract $\frac{1}{2}$ from $0$ .

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

Step 9.3.10

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

$\frac{1}{3} - \frac{1}{3 \cdot 2}$

Step 9.3.11

Multiply $3$ by $2$ .

$\frac{1}{3} - \frac{1}{6}$

Step 9.3.12

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{3} \cdot \frac{2}{2} - \frac{1}{6}$

Step 9.3.13

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 9.3.13.1

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

$\frac{2}{3 \cdot 2} - \frac{1}{6}$

Step 9.3.13.2

Multiply $3$ by $2$ .

$\frac{2}{6} - \frac{1}{6}$

Step 9.3.14

Combine the numerators over the common denominator.

$\frac{2 - 1}{6}$

Step 9.3.15

Subtract $1$ from $2$ .

$\frac{1}{6}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$