Calculus Examples

\int t e^{- 3 t} d t

$\int t e^{- 3 t} d t$

Step 1

Integrate by parts using the formula

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , where

u = t

$u = t$ and

d v = e^{- 3 t}

$d v = e^{- 3 t}$ .

t (- \frac{1}{3} e^{- 3 t}) - \int - \frac{1}{3} e^{- 3 t} d t

$t (- \frac{1}{3} e^{- 3 t}) - \int - \frac{1}{3} e^{- 3 t} d t$

Step 2

Simplify.

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Step 2.1

Combine

e^{- 3 t}

$e^{- 3 t}$ and

\frac{1}{3}

$\frac{1}{3}$ .

t (- \frac{e^{- 3 t}}{3}) - \int - \frac{1}{3} e^{- 3 t} d t

$t (- \frac{e^{- 3 t}}{3}) - \int - \frac{1}{3} e^{- 3 t} d t$

Step 2.2

Combine

t

$t$ and

\frac{e^{- 3 t}}{3}

$\frac{e^{- 3 t}}{3}$ .

- \frac{t e^{- 3 t}}{3} - \int - \frac{1}{3} e^{- 3 t} d t

$- \frac{t e^{- 3 t}}{3} - \int - \frac{1}{3} e^{- 3 t} d t$

- \frac{t e^{- 3 t}}{3} - \int - \frac{1}{3} e^{- 3 t} d t

$- \frac{t e^{- 3 t}}{3} - \int - \frac{1}{3} e^{- 3 t} d t$

Step 3

Since

- \frac{1}{3}

$- \frac{1}{3}$ is constant with respect to

t

$t$ , move

- \frac{1}{3}

$- \frac{1}{3}$ out of the integral.

- \frac{t e^{- 3 t}}{3} - (- \frac{1}{3} \int e^{- 3 t} d t)

$- \frac{t e^{- 3 t}}{3} - (- \frac{1}{3} \int e^{- 3 t} d t)$

Step 4

Simplify.

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Step 4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

- \frac{t e^{- 3 t}}{3} + 1 (\frac{1}{3} \int e^{- 3 t} d t)

$- \frac{t e^{- 3 t}}{3} + 1 (\frac{1}{3} \int e^{- 3 t} d t)$

Step 4.2

Multiply

\frac{1}{3}

$\frac{1}{3}$ by

1

$1$ .

- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int e^{- 3 t} d t

$- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int e^{- 3 t} d t$

- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int e^{- 3 t} d t

$- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int e^{- 3 t} d t$

Step 5

Let

u = - 3 t

$u = - 3 t$ . Then

d u = - 3 d t

$d u = - 3 d t$ , so

- \frac{1}{3} d u = d t

$- \frac{1}{3} d u = d t$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 5.1

Let

u = - 3 t

$u = - 3 t$ . Find

\frac{d u}{d t}

$\frac{d u}{d t}$ .

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Step 5.1.1

Differentiate

- 3 t

$- 3 t$ .

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

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

Step 5.1.2

Since

- 3

$- 3$ is constant with respect to

t

$t$ , the derivative of

- 3 t

$- 3 t$ with respect to

t

$t$ is

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

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

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

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

Step 5.1.3

Differentiate using the Power Rule which states that

\frac{d}{d t} [t^{n}]

$\frac{d}{d t} [t^{n}]$ is

n t^{n - 1}

$n t^{n - 1}$ where

n = 1

$n = 1$ .

- 3 \cdot 1

$- 3 \cdot 1$

Step 5.1.4

Multiply

- 3

$- 3$ by

1

$1$ .

- 3

$- 3$

- 3

$- 3$

Step 5.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int e^{u} \frac{1}{- 3} d u

$- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int e^{u} \frac{1}{- 3} d u$

- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int e^{u} \frac{1}{- 3} d u

$- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int e^{u} \frac{1}{- 3} d u$

Step 6

Simplify.

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Step 6.1

Move the negative in front of the fraction.

- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int e^{u} (- \frac{1}{3}) d u

$- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int e^{u} (- \frac{1}{3}) d u$

Step 6.2

Combine

e^{u}

$e^{u}$ and

\frac{1}{3}

$\frac{1}{3}$ .

- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int - \frac{e^{u}}{3} d u

$- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int - \frac{e^{u}}{3} d u$

- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int - \frac{e^{u}}{3} d u

$- \frac{t e^{- 3 t}}{3} + \frac{1}{3} \int - \frac{e^{u}}{3} d u$

Step 7

Since

- 1

$- 1$ is constant with respect to

u

$u$ , move

- 1

$- 1$ out of the integral.

- \frac{t e^{- 3 t}}{3} + \frac{1}{3} (- \int \frac{e^{u}}{3} d u)

$- \frac{t e^{- 3 t}}{3} + \frac{1}{3} (- \int \frac{e^{u}}{3} d u)$

Step 8

Since

\frac{1}{3}

$\frac{1}{3}$ is constant with respect to

u

$u$ , move

\frac{1}{3}

$\frac{1}{3}$ out of the integral.

- \frac{t e^{- 3 t}}{3} + \frac{1}{3} (- (\frac{1}{3} \int e^{u} d u))

$- \frac{t e^{- 3 t}}{3} + \frac{1}{3} (- (\frac{1}{3} \int e^{u} d u))$

Step 9

Simplify.

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Step 9.1

Multiply

\frac{1}{3}

$\frac{1}{3}$ by

\frac{1}{3}

$\frac{1}{3}$ .

- \frac{t e^{- 3 t}}{3} - \frac{1}{3 \cdot 3} \int e^{u} d u

$- \frac{t e^{- 3 t}}{3} - \frac{1}{3 \cdot 3} \int e^{u} d u$

Step 9.2

Multiply

3

$3$ by

3

$3$ .

- \frac{t e^{- 3 t}}{3} - \frac{1}{9} \int e^{u} d u

$- \frac{t e^{- 3 t}}{3} - \frac{1}{9} \int e^{u} d u$

- \frac{t e^{- 3 t}}{3} - \frac{1}{9} \int e^{u} d u

$- \frac{t e^{- 3 t}}{3} - \frac{1}{9} \int e^{u} d u$

Step 10

The integral of

e^{u}

$e^{u}$ with respect to

u

$u$ is

e^{u}

$e^{u}$ .

- \frac{t e^{- 3 t}}{3} - \frac{1}{9} (e^{u} + C)

$- \frac{t e^{- 3 t}}{3} - \frac{1}{9} (e^{u} + C)$

Step 11

Simplify.

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Step 11.1

Rewrite

- \frac{t e^{- 3 t}}{3} - \frac{1}{9} (e^{u} + C)

$- \frac{t e^{- 3 t}}{3} - \frac{1}{9} (e^{u} + C)$ as

- \frac{1}{3} t e^{- 3 t} - \frac{1}{9} e^{u} + C

$- \frac{1}{3} t e^{- 3 t} - \frac{1}{9} e^{u} + C$ .

- \frac{1}{3} t e^{- 3 t} - \frac{1}{9} e^{u} + C

$- \frac{1}{3} t e^{- 3 t} - \frac{1}{9} e^{u} + C$

Step 11.2

Simplify.

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Step 11.2.1

Combine

t

$t$ and

\frac{1}{3}

$\frac{1}{3}$ .

- \frac{t}{3} e^{- 3 t} - \frac{1}{9} e^{u} + C

$- \frac{t}{3} e^{- 3 t} - \frac{1}{9} e^{u} + C$

Step 11.2.2

Combine

e^{- 3 t}

$e^{- 3 t}$ and

\frac{t}{3}

$\frac{t}{3}$ .

- \frac{e^{- 3 t} t}{3} - \frac{1}{9} e^{u} + C

$- \frac{e^{- 3 t} t}{3} - \frac{1}{9} e^{u} + C$

- \frac{e^{- 3 t} t}{3} - \frac{1}{9} e^{u} + C

$- \frac{e^{- 3 t} t}{3} - \frac{1}{9} e^{u} + C$

- \frac{e^{- 3 t} t}{3} - \frac{1}{9} e^{u} + C

$- \frac{e^{- 3 t} t}{3} - \frac{1}{9} e^{u} + C$

Step 12

Replace all occurrences of

u

$u$ with

- 3 t

$- 3 t$ .

- \frac{e^{- 3 t} t}{3} - \frac{1}{9} e^{- 3 t} + C

$- \frac{e^{- 3 t} t}{3} - \frac{1}{9} e^{- 3 t} + C$

Step 13

Combine

e^{- 3 t}

$e^{- 3 t}$ and

\frac{1}{9}

$\frac{1}{9}$ .

- \frac{e^{- 3 t} t}{3} - \frac{e^{- 3 t}}{9} + C

$- \frac{e^{- 3 t} t}{3} - \frac{e^{- 3 t}}{9} + C$

Step 14

Reorder terms.

- \frac{1}{3} e^{- 3 t} t - \frac{1}{9} e^{- 3 t} + C

$- \frac{1}{3} e^{- 3 t} t - \frac{1}{9} e^{- 3 t} + C$