Calculus Examples

$\int_{0}^{3} e^{t} - e^{- t} d t$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{3} e^{t} d t + \int_{0}^{3} - e^{- t} d t$

Step 2

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

$e^{t}]_{0}^{3} + \int_{0}^{3} - e^{- t} d t$

Step 3

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

$e^{t}]_{0}^{3} - \int_{0}^{3} e^{- t} d t$

Step 4

Let $u = - t$ . Then $d u = - d t$ , so $- d u = d t$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

Differentiate $- t$ .

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

Step 4.1.2

Since $- 1$ is constant with respect to $t$ , the derivative of $- t$ with respect to $t$ is $- \frac{d}{d t} [t]$ .

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

Step 4.1.3

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

$- 1 \cdot 1$

Step 4.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.2

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

$u_{lower} = - 0$

Step 4.3

Multiply $- 1$ by $0$ .

$u_{lower} = 0$

Step 4.4

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

$u_{upper} = - 1 \cdot 3$

Step 4.5

Multiply $- 1$ by $3$ .

$u_{upper} = - 3$

Step 4.6

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

$u_{lower} = 0$

$u_{upper} = - 3$

Step 4.7

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

$e^{t}]_{0}^{3} - \int_{0}^{- 3} - e^{u} d u$

Step 5

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

$e^{t}]_{0}^{3} - - \int_{0}^{- 3} e^{u} d u$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $- 1$ .

$e^{t}]_{0}^{3} + 1 \int_{0}^{- 3} e^{u} d u$

Step 6.2

Multiply $\int_{0}^{- 3} e^{u} d u$ by $1$ .

$e^{t}]_{0}^{3} + \int_{0}^{- 3} e^{u} d u$

Step 7

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

$e^{t}]_{0}^{3} + e^{u}]_{0}^{- 3}$

Step 8

Substitute and simplify.

Tap for more steps...

Step 8.1

Evaluate $e^{t}$ at $3$ and at $0$ .

$(e^{3}) - e^{0} + e^{u}]_{0}^{- 3}$

Step 8.2

Evaluate $e^{u}$ at $- 3$ and at $0$ .

$(e^{3}) - e^{0} + (e^{- 3}) - e^{0}$

Step 8.3

Simplify.

Tap for more steps...

Step 8.3.1

Anything raised to $0$ is $1$ .

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

Step 8.3.2

Multiply $- 1$ by $1$ .

$e^{3} - 1 + (e^{- 3}) - e^{0}$

Step 8.3.3

Anything raised to $0$ is $1$ .

$e^{3} - 1 + e^{- 3} - 1 \cdot 1$

Step 8.3.4

Multiply $- 1$ by $1$ .

$e^{3} - 1 + e^{- 3} - 1$

Step 8.3.5

Subtract $1$ from $- 1$ .

$e^{3} + e^{- 3} - 2$

Step 9

The result can be shown in multiple forms.

Exact Form:

$e^{3} + e^{- 3} - 2$

Decimal Form:

$18.13532399 \dots$

Step 10