Calculus Examples

$\int t e^{- 8 t} d t$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t$ and $d v = e^{- 8 t}$ .

$t (- \frac{1}{8} e^{- 8 t}) - \int - \frac{1}{8} e^{- 8 t} d t$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Combine $e^{- 8 t}$ and $\frac{1}{8}$ .

$t (- \frac{e^{- 8 t}}{8}) - \int - \frac{1}{8} e^{- 8 t} d t$

Step 2.2

Combine $t$ and $\frac{e^{- 8 t}}{8}$ .

$- \frac{t e^{- 8 t}}{8} - \int - \frac{1}{8} e^{- 8 t} d t$

Step 3

Since $- \frac{1}{8}$ is constant with respect to $t$ , move $- \frac{1}{8}$ out of the integral.

$- \frac{t e^{- 8 t}}{8} - (- \frac{1}{8} \int e^{- 8 t} d t)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Multiply $- 1$ by $- 1$ .

$- \frac{t e^{- 8 t}}{8} + 1 (\frac{1}{8} \int e^{- 8 t} d t)$

Step 4.2

Multiply $\frac{1}{8}$ by $1$ .

$- \frac{t e^{- 8 t}}{8} + \frac{1}{8} \int e^{- 8 t} d t$

Step 5

Let $u = - 8 t$ . Then $d u = - 8 d t$ , so $- \frac{1}{8} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 5.1

Let $u = - 8 t$ . Find $\frac{d u}{d t}$ .

Tap for more steps...

Step 5.1.1

Differentiate $- 8 t$ .

$\frac{d}{d t} [- 8 t]$

Step 5.1.2

Since $- 8$ is constant with respect to $t$ , the derivative of $- 8 t$ with respect to $t$ is $- 8 \frac{d}{d t} [t]$ .

$- 8 \frac{d}{d t} [t]$

Step 5.1.3

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

$- 8 \cdot 1$

Step 5.1.4

Multiply $- 8$ by $1$ .

$- 8$

Step 5.2

Rewrite the problem using $u$ and $d u$ .

$- \frac{t e^{- 8 t}}{8} + \frac{1}{8} \int e^{u} \frac{1}{- 8} d u$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Move the negative in front of the fraction.

$- \frac{t e^{- 8 t}}{8} + \frac{1}{8} \int e^{u} (- \frac{1}{8}) d u$

Step 6.2

Combine $e^{u}$ and $\frac{1}{8}$ .

$- \frac{t e^{- 8 t}}{8} + \frac{1}{8} \int - \frac{e^{u}}{8} d u$

Step 7

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

$- \frac{t e^{- 8 t}}{8} + \frac{1}{8} (- \int \frac{e^{u}}{8} d u)$

Step 8

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$- \frac{t e^{- 8 t}}{8} + \frac{1}{8} (- (\frac{1}{8} \int e^{u} d u))$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Multiply $\frac{1}{8}$ by $\frac{1}{8}$ .

$- \frac{t e^{- 8 t}}{8} - \frac{1}{8 \cdot 8} \int e^{u} d u$

Step 9.2

Multiply $8$ by $8$ .

$- \frac{t e^{- 8 t}}{8} - \frac{1}{64} \int e^{u} d u$

Step 10

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$- \frac{t e^{- 8 t}}{8} - \frac{1}{64} (e^{u} + C)$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Rewrite $- \frac{t e^{- 8 t}}{8} - \frac{1}{64} (e^{u} + C)$ as $- \frac{1}{8} t e^{- 8 t} - \frac{1}{64} e^{u} + C$ .

$- \frac{1}{8} t e^{- 8 t} - \frac{1}{64} e^{u} + C$

Step 11.2

Simplify.

Tap for more steps...

Step 11.2.1

Combine $t$ and $\frac{1}{8}$ .

$- \frac{t}{8} e^{- 8 t} - \frac{1}{64} e^{u} + C$

Step 11.2.2

Combine $e^{- 8 t}$ and $\frac{t}{8}$ .

$- \frac{e^{- 8 t} t}{8} - \frac{1}{64} e^{u} + C$

Step 12

Replace all occurrences of $u$ with $- 8 t$ .

$- \frac{e^{- 8 t} t}{8} - \frac{1}{64} e^{- 8 t} + C$

Step 13

Combine $e^{- 8 t}$ and $\frac{1}{64}$ .

$- \frac{e^{- 8 t} t}{8} - \frac{e^{- 8 t}}{64} + C$

Step 14

Reorder terms.

$- \frac{1}{8} e^{- 8 t} t - \frac{1}{64} e^{- 8 t} + C$