Calculus Examples

$\int (t^{2} + t) \cos (3 t) d t$

Step 1

Apply the distributive property.

$\int t^{2} \cos (3 t) + t \cos (3 t) d t$

Step 2

Split the single integral into multiple integrals.

$\int t^{2} \cos (3 t) d t + \int t \cos (3 t) d t$

Step 3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t^{2}$ and $d v = \cos (3 t)$ .

$t^{2} (\frac{1}{3} \sin (3 t)) - \int \frac{1}{3} \sin (3 t) (2 t) d t + \int t \cos (3 t) d t$

Step 4

Simplify.

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Step 4.1

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

$t^{2} \frac{\sin (3 t)}{3} - \int \frac{1}{3} \sin (3 t) (2 t) d t + \int t \cos (3 t) d t$

Step 4.2

Combine $t^{2}$ and $\frac{\sin (3 t)}{3}$ .

$\frac{t^{2} \sin (3 t)}{3} - \int \frac{1}{3} \sin (3 t) (2 t) d t + \int t \cos (3 t) d t$

Step 5

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

$\frac{t^{2} \sin (3 t)}{3} - (\frac{1}{3} \cdot 2 \int \sin (3 t) (t) d t) + \int t \cos (3 t) d t$

Step 6

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

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} \int \sin (3 t) (t) d t + \int t \cos (3 t) d t$

Step 7

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t$ and $d v = \sin (3 t)$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (t (- \frac{1}{3} \cos (3 t)) - \int - \frac{1}{3} \cos (3 t) d t) + \int t \cos (3 t) d t$

Step 8

Simplify.

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Step 8.1

Combine $\cos (3 t)$ and $\frac{1}{3}$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (t (- \frac{\cos (3 t)}{3}) - \int - \frac{1}{3} \cos (3 t) d t) + \int t \cos (3 t) d t$

Step 8.2

Combine $t$ and $\frac{\cos (3 t)}{3}$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} - \int - \frac{1}{3} \cos (3 t) d t) + \int t \cos (3 t) d t$

Step 8.3

Combine $\cos (3 t)$ and $\frac{1}{3}$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} - \int - \frac{\cos (3 t)}{3} d t) + \int t \cos (3 t) d t$

Step 9

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} - - \int \frac{\cos (3 t)}{3} d t) + \int t \cos (3 t) d t$

Step 10

Simplify.

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Step 10.1

Multiply $- 1$ by $- 1$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + 1 \int \frac{\cos (3 t)}{3} d t) + \int t \cos (3 t) d t$

Step 10.2

Multiply $\int \frac{\cos (3 t)}{3} d t$ by $1$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \int \frac{\cos (3 t)}{3} d t) + \int t \cos (3 t) d t$

Step 11

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

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{3} \int \cos (3 t) d t) + \int t \cos (3 t) d t$

Step 12

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

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Step 12.1

Let $u_{1} = 3 t$ . Find $\frac{d u_{1}}{d t}$ .

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Step 12.1.1

Differentiate $3 t$ .

$\frac{d}{d t} [3 t]$

Step 12.1.2

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

$3 \frac{d}{d t} [t]$

Step 12.1.3

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

$3 \cdot 1$

Step 12.1.4

Multiply $3$ by $1$ .

$3$

Step 12.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{3} \int \cos (u_{1}) \frac{1}{3} d u_{1}) + \int t \cos (3 t) d t$

Step 13

Combine $\cos (u_{1})$ and $\frac{1}{3}$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{3} \int \frac{\cos (u_{1})}{3} d u_{1}) + \int t \cos (3 t) d t$

Step 14

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

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{3} (\frac{1}{3} \int \cos (u_{1}) d u_{1})) + \int t \cos (3 t) d t$

Step 15

Simplify.

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Step 15.1

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

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{3 \cdot 3} \int \cos (u_{1}) d u_{1}) + \int t \cos (3 t) d t$

Step 15.2

Multiply $3$ by $3$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} \int \cos (u_{1}) d u_{1}) + \int t \cos (3 t) d t$

Step 16

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

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} (\sin (u_{1}) + C)) + \int t \cos (3 t) d t$

Step 17

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t$ and $d v = \cos (3 t)$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} (\sin (u_{1}) + C)) + t (\frac{1}{3} \sin (3 t)) - \int \frac{1}{3} \sin (3 t) d t$

Step 18

Simplify.

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Step 18.1

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

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} (\sin (u_{1}) + C)) + t \frac{\sin (3 t)}{3} - \int \frac{1}{3} \sin (3 t) d t$

Step 18.2

Combine $t$ and $\frac{\sin (3 t)}{3}$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} (\sin (u_{1}) + C)) + \frac{t \sin (3 t)}{3} - \int \frac{1}{3} \sin (3 t) d t$

Step 19

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

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} (\sin (u_{1}) + C)) + \frac{t \sin (3 t)}{3} - (\frac{1}{3} \int \sin (3 t) d t)$

Step 20

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

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Step 20.1

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

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Step 20.1.1

Differentiate $3 t$ .

$\frac{d}{d t} [3 t]$

Step 20.1.2

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

$3 \frac{d}{d t} [t]$

Step 20.1.3

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

$3 \cdot 1$

Step 20.1.4

Multiply $3$ by $1$ .

$3$

Step 20.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} (\sin (u_{1}) + C)) + \frac{t \sin (3 t)}{3} - \frac{1}{3} \int \sin (u_{2}) \frac{1}{3} d u_{2}$

Step 21

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

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} (\sin (u_{1}) + C)) + \frac{t \sin (3 t)}{3} - \frac{1}{3} \int \frac{\sin (u_{2})}{3} d u_{2}$

Step 22

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

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} (\sin (u_{1}) + C)) + \frac{t \sin (3 t)}{3} - \frac{1}{3} (\frac{1}{3} \int \sin (u_{2}) d u_{2})$

Step 23

Simplify.

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Step 23.1

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

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} (\sin (u_{1}) + C)) + \frac{t \sin (3 t)}{3} - \frac{1}{3 \cdot 3} \int \sin (u_{2}) d u_{2}$

Step 23.2

Multiply $3$ by $3$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} (\sin (u_{1}) + C)) + \frac{t \sin (3 t)}{3} - \frac{1}{9} \int \sin (u_{2}) d u_{2}$

Step 24

The integral of $\sin (u_{2})$ with respect to $u_{2}$ is $- \cos (u_{2})$ .

$\frac{t^{2} \sin (3 t)}{3} - \frac{2}{3} (- \frac{t \cos (3 t)}{3} + \frac{1}{9} (\sin (u_{1}) + C)) + \frac{t \sin (3 t)}{3} - \frac{1}{9} (- \cos (u_{2}) + C)$

Step 25

Simplify.

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Step 25.1

Simplify.

$\frac{t^{2} \sin (3 t)}{3} + \frac{2 t \cos (3 t)}{9} - \frac{2 \sin (u_{1})}{27} + \frac{t \sin (3 t)}{3} - \frac{1}{9} (- \cos (u_{2})) + C$

Step 25.2

Simplify.

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Step 25.2.1

Multiply $- 1$ by $- 1$ .

$\frac{t^{2} \sin (3 t)}{3} + \frac{2 t \cos (3 t)}{9} - \frac{2 \sin (u_{1})}{27} + \frac{t \sin (3 t)}{3} + 1 (\frac{1}{9}) \cos (u_{2}) + C$

Step 25.2.2

Multiply $\frac{1}{9}$ by $1$ .

$\frac{t^{2} \sin (3 t)}{3} + \frac{2 t \cos (3 t)}{9} - \frac{2 \sin (u_{1})}{27} + \frac{t \sin (3 t)}{3} + \frac{1}{9} \cos (u_{2}) + C$

Step 25.2.3

Combine $\frac{1}{9}$ and $\cos (u_{2})$ .

$\frac{t^{2} \sin (3 t)}{3} + \frac{2 t \cos (3 t)}{9} - \frac{2 \sin (u_{1})}{27} + \frac{t \sin (3 t)}{3} + \frac{\cos (u_{2})}{9} + C$

Step 26

Substitute back in for each integration substitution variable.

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Step 26.1

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

$\frac{t^{2} \sin (3 t)}{3} + \frac{2 t \cos (3 t)}{9} - \frac{2 \sin (3 t)}{27} + \frac{t \sin (3 t)}{3} + \frac{\cos (u_{2})}{9} + C$

Step 26.2

Replace all occurrences of $u_{2}$ with $3 t$ .

$\frac{t^{2} \sin (3 t)}{3} + \frac{2 t \cos (3 t)}{9} - \frac{2 \sin (3 t)}{27} + \frac{t \sin (3 t)}{3} + \frac{\cos (3 t)}{9} + C$

Step 27

Reorder terms.

$\frac{1}{3} t^{2} \sin (3 t) + \frac{2}{9} t \cos (3 t) - \frac{2}{27} \sin (3 t) + \frac{1}{3} t \sin (3 t) + \frac{1}{9} \cos (3 t) + C$