Calculus Examples

$\int_{0}^{2 π} t^{2} \sin (2 t) d t$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t^{2}$ and $d v = \sin (2 t)$ .

$t^{2} (- \frac{1}{2} \cos (2 t))]_{0}^{2 π} - \int_{0}^{2 π} - \frac{1}{2} \cos (2 t) (2 t) d t$

Step 2

Simplify.

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Step 2.1

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

$t^{2} (- \frac{\cos (2 t)}{2})]_{0}^{2 π} - \int_{0}^{2 π} - \frac{1}{2} \cos (2 t) (2 t) d t$

Step 2.2

Combine $t^{2}$ and $\frac{\cos (2 t)}{2}$ .

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} - \int_{0}^{2 π} - \frac{1}{2} \cos (2 t) (2 t) d t$

Step 3

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

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} - (- \frac{1}{2} \cdot 2 \int_{0}^{2 π} \cos (2 t) (t) d t)$

Step 4

Simplify.

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Step 4.1

Multiply $2$ by $- 1$ .

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} - (- 2 (\frac{1}{2}) \int_{0}^{2 π} \cos (2 t) (t) d t)$

Step 4.2

Combine $- 2$ and $\frac{1}{2}$ .

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} - (\frac{- 2}{2} \int_{0}^{2 π} \cos (2 t) (t) d t)$

Step 4.3

Cancel the common factor of $- 2$ and $2$ .

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Step 4.3.1

Factor $2$ out of $- 2$ .

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} - (\frac{2 \cdot - 1}{2} \int_{0}^{2 π} \cos (2 t) (t) d t)$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $2$ .

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} - (\frac{2 \cdot - 1}{2 (1)} \int_{0}^{2 π} \cos (2 t) (t) d t)$

Step 4.3.2.2

Cancel the common factor.

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} - (\frac{2 \cdot - 1}{2 \cdot 1} \int_{0}^{2 π} \cos (2 t) (t) d t)$

Step 4.3.2.3

Rewrite the expression.

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} - (\frac{- 1}{1} \int_{0}^{2 π} \cos (2 t) (t) d t)$

Step 4.3.2.4

Divide $- 1$ by $1$ .

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} - - \int_{0}^{2 π} \cos (2 t) (t) d t$

Step 4.4

Multiply $- 1$ by $- 1$ .

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + 1 \int_{0}^{2 π} \cos (2 t) (t) d t$

Step 4.5

Multiply $\int_{0}^{2 π} \cos (2 t) (t) d t$ by $1$ .

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + \int_{0}^{2 π} \cos (2 t) (t) d t$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t$ and $d v = \cos (2 t)$ .

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + t (\frac{1}{2} \sin (2 t))]_{0}^{2 π} - \int_{0}^{2 π} \frac{1}{2} \sin (2 t) d t$

Step 6

Simplify.

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Step 6.1

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

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + t \frac{\sin (2 t)}{2}]_{0}^{2 π} - \int_{0}^{2 π} \frac{1}{2} \sin (2 t) d t$

Step 6.2

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

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + \frac{t \sin (2 t)}{2}]_{0}^{2 π} - \int_{0}^{2 π} \frac{1}{2} \sin (2 t) d t$

Step 7

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

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + \frac{t \sin (2 t)}{2}]_{0}^{2 π} - (\frac{1}{2} \int_{0}^{2 π} \sin (2 t) d t)$

Step 8

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

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Step 8.1

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

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Step 8.1.1

Differentiate $2 t$ .

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

Step 8.1.2

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

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

Step 8.1.3

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

$2 \cdot 1$

Step 8.1.4

Multiply $2$ by $1$ .

$2$

Step 8.2

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

$u_{lower} = 2 \cdot 0$

Step 8.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 8.4

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

$u_{upper} = 2 (2 π)$

Step 8.5

Multiply $2$ by $2$ .

$u_{upper} = 4 π$

Step 8.6

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

$u_{lower} = 0$

$u_{upper} = 4 π$

Step 8.7

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

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + \frac{t \sin (2 t)}{2}]_{0}^{2 π} - \frac{1}{2} \int_{0}^{4 π} \sin (u) \frac{1}{2} d u$

Step 9

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

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + \frac{t \sin (2 t)}{2}]_{0}^{2 π} - \frac{1}{2} \int_{0}^{4 π} \frac{\sin (u)}{2} d u$

Step 10

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

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + \frac{t \sin (2 t)}{2}]_{0}^{2 π} - \frac{1}{2} (\frac{1}{2} \int_{0}^{4 π} \sin (u) d u)$

Step 11

Simplify.

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Step 11.1

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

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + \frac{t \sin (2 t)}{2}]_{0}^{2 π} - \frac{1}{2 \cdot 2} \int_{0}^{4 π} \sin (u) d u$

Step 11.2

Multiply $2$ by $2$ .

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + \frac{t \sin (2 t)}{2}]_{0}^{2 π} - \frac{1}{4} \int_{0}^{4 π} \sin (u) d u$

Step 12

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

$- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π} + \frac{t \sin (2 t)}{2}]_{0}^{2 π} - \frac{1}{4} - \cos (u)]_{0}^{4 π}$

Step 13

Reduce the expression by cancelling the common factors.

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Step 13.1

Combine $- \frac{t^{2} \cos (2 t)}{2}]_{0}^{2 π}$ and $\frac{t \sin (2 t)}{2}]_{0}^{2 π}$ .

$- \frac{t^{2} \cos (2 t)}{2} + \frac{t \sin (2 t)}{2}]_{0}^{2 π} - \frac{1}{4} - \cos (u)]_{0}^{4 π}$

Step 13.2

Simplify the expression.

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Step 13.2.1

Insert parentheses.

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Step 13.2.1.1

Evaluate $- \frac{t^{2} \cos (2 t)}{2} + \frac{t \sin (2 t)}{2}$ at $2 π$ and at $0$ .

$(- \frac{{(2 π)}^{2} \cos (2 (2 π))}{2} + \frac{2 π \sin (2 (2 π))}{2}) - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} - \cos (u)]_{0}^{4 π}$

Step 13.2.1.2

Evaluate $- \cos (u)$ at $4 π$ and at $0$ .

$(- \frac{{(2 π)}^{2} \cos (2 (2 π))}{2} + \frac{2 π \sin (2 (2 π))}{2}) - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.2.2

Simplify.

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Step 13.2.2.1

Factor $2$ out of $2 π$ .

$- \frac{{(2 (π))}^{2} \cos (2 (2 π))}{2} + \frac{2 π \sin (2 (2 π))}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.2.2.2

Apply the product rule to $2 (π)$ .

$- \frac{2^{2} π^{2} \cos (2 (2 π))}{2} + \frac{2 π \sin (2 (2 π))}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.2.2.3

Raise $2$ to the power of $2$ .

$- \frac{4 π^{2} \cos (2 (2 π))}{2} + \frac{2 π \sin (2 (2 π))}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.2.3

Multiply $2$ by $2$ .

$- \frac{4 π^{2} \cos (4 π)}{2} + \frac{2 π \sin (2 (2 π))}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.3

Cancel the common factor of $4$ and $2$ .

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Step 13.3.1

Factor $2$ out of $4 π^{2} \cos (4 π)$ .

$- \frac{2 (2 π^{2} \cos (4 π))}{2} + \frac{2 π \sin (2 (2 π))}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.3.2

Cancel the common factors.

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Step 13.3.2.1

Factor $2$ out of $2$ .

$- \frac{2 (2 π^{2} \cos (4 π))}{2 (1)} + \frac{2 π \sin (2 (2 π))}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.3.2.2

Cancel the common factor.

$- \frac{2 (2 π^{2} \cos (4 π))}{2 \cdot 1} + \frac{2 π \sin (2 (2 π))}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.3.2.3

Rewrite the expression.

$- \frac{2 π^{2} \cos (4 π)}{1} + \frac{2 π \sin (2 (2 π))}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.3.2.4

Divide $2 π^{2} \cos (4 π)$ by $1$ .

$- (2 π^{2} \cos (4 π)) + \frac{2 π \sin (2 (2 π))}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.4

Simplify the expression.

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Step 13.4.1

Multiply $2$ by $- 1$ .

$- 2 π^{2} \cos (4 π) + \frac{2 π \sin (2 (2 π))}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.4.2

Multiply $2$ by $2$ .

$- 2 π^{2} \cos (4 π) + \frac{2 π \sin (4 π)}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.5

Cancel the common factor of $2$ .

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Step 13.5.1

Cancel the common factor.

$- 2 π^{2} \cos (4 π) + \frac{2 π \sin (4 π)}{2} - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.5.2

Divide $π \sin (4 π)$ by $1$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (- \frac{0^{2} \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.6

Simplify the expression.

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Step 13.6.1

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

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (- \frac{0 \cos (2 \cdot 0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.6.2

Multiply $2$ by $0$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (- \frac{0 \cos (0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.6.3

Multiply $0$ by $\cos (0)$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (- \frac{0}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.7

Cancel the common factor of $0$ and $2$ .

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Step 13.7.1

Factor $2$ out of $0$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (- \frac{2 (0)}{2} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.7.2

Cancel the common factors.

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Step 13.7.2.1

Factor $2$ out of $2$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (- \frac{2 \cdot 0}{2 \cdot 1} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.7.2.2

Cancel the common factor.

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (- \frac{2 \cdot 0}{2 \cdot 1} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.7.2.3

Rewrite the expression.

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (- \frac{0}{1} + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.7.2.4

Divide $0$ by $1$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (- 0 + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.8

Multiply by zero.

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Step 13.8.1

Multiply $- 1$ by $0$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (0 + \frac{0 \sin (2 \cdot 0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.8.2

Multiply $2$ by $0$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (0 + \frac{0 \sin (0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.8.3

Multiply $0$ by $\sin (0)$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (0 + \frac{0}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.9

Cancel the common factor of $0$ and $2$ .

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Step 13.9.1

Factor $2$ out of $0$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (0 + \frac{2 (0)}{2}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.9.2

Cancel the common factors.

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Step 13.9.2.1

Factor $2$ out of $2$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (0 + \frac{2 \cdot 0}{2 \cdot 1}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.9.2.2

Cancel the common factor.

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (0 + \frac{2 \cdot 0}{2 \cdot 1}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.9.2.3

Rewrite the expression.

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (0 + \frac{0}{1}) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.9.2.4

Divide $0$ by $1$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - (0 + 0) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.10

Simplify the expression.

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Step 13.10.1

Add $0$ and $0$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - 0 - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.10.2

Multiply $- 1$ by $0$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) + 0 - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 13.10.3

Add $- 2 π^{2} \cos (4 π) + π \sin (4 π)$ and $0$ .

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - \frac{1}{4} ((- \cos (4 π)) + \cos (0))$

Step 14

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

$- 2 π^{2} \cos (4 π) + π \sin (4 π) - \frac{1}{4} (- \cos (4 π) + 1)$

Step 15

Simplify.

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Step 15.1

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

$- 2 π^{2} \cos (0) + π \sin (4 π) - \frac{1}{4} (- \cos (4 π) + 1)$

Step 15.2

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

$- 2 π^{2} \cdot 1 + π \sin (4 π) - \frac{1}{4} (- \cos (4 π) + 1)$

Step 15.3

Multiply $- 2$ by $1$ .

$- 2 π^{2} + π \sin (4 π) - \frac{1}{4} (- \cos (4 π) + 1)$

Step 15.4

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

$- 2 π^{2} + π \sin (0) - \frac{1}{4} (- \cos (4 π) + 1)$

Step 15.5

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

$- 2 π^{2} + π \cdot 0 - \frac{1}{4} (- \cos (4 π) + 1)$

Step 15.6

Multiply $π$ by $0$ .

$- 2 π^{2} + 0 - \frac{1}{4} (- \cos (4 π) + 1)$

Step 15.7

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

$- 2 π^{2} + 0 - \frac{1}{4} (- \cos (0) + 1)$

Step 15.8

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

$- 2 π^{2} + 0 - \frac{1}{4} (- 1 \cdot 1 + 1)$

Step 15.9

Multiply $- 1$ by $1$ .

$- 2 π^{2} + 0 - \frac{1}{4} (- 1 + 1)$

Step 15.10

Add $- 1$ and $1$ .

$- 2 π^{2} + 0 - \frac{1}{4} \cdot 0$

Step 15.11

Multiply $- \frac{1}{4} \cdot 0$ .

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Step 15.11.1

Multiply $0$ by $- 1$ .

$- 2 π^{2} + 0 + 0 (\frac{1}{4})$

Step 15.11.2

Multiply $0$ by $\frac{1}{4}$ .

$- 2 π^{2} + 0 + 0$

Step 15.12

Add $- 2 π^{2}$ and $0$ .

$- 2 π^{2} + 0$

Step 15.13

Add $- 2 π^{2}$ and $0$ .

$- 2 π^{2}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- 2 π^{2}$

Decimal Form:

$- 19.73920880 \dots$