Calculus Examples

$\int_{- 1}^{3} e^{u + 1} d u$

Step 1

Let $u_{1} = u + 1$ . Then $d u_{1} = d u$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = u + 1$ . Find $\frac{d u_{1}}{d u}$ .

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Step 1.1.1

Differentiate $u + 1$ .

$\frac{d}{d u} [u + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $u + 1$ with respect to $u$ is $\frac{d}{d u} [u] + \frac{d}{d u} [1]$ .

$\frac{d}{d u} [u] + \frac{d}{d u} [1]$

Step 1.1.3

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

$1 + \frac{d}{d u} [1]$

Step 1.1.4

Since $1$ is constant with respect to $u$ , the derivative of $1$ with respect to $u$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

Substitute the lower limit in for $u$ in $u_{1} = u + 1$ .

$u_{lower} = - 1 + 1$

Step 1.3

Add $- 1$ and $1$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $u$ in $u_{1} = u + 1$ .

$u_{upper} = 3 + 1$

Step 1.5

Add $3$ and $1$ .

$u_{upper} = 4$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 4$

Step 1.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$\int_{0}^{4} e^{u_{1}} d u_{1}$

Step 2

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

$e^{u_{1}}]_{0}^{4}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $e^{u_{1}}$ at $4$ and at $0$ .

$(e^{4}) - e^{0}$

Step 3.2

Simplify.

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Step 3.2.1

Anything raised to $0$ is $1$ .

$e^{4} - 1 \cdot 1$

Step 3.2.2

Multiply $- 1$ by $1$ .

$e^{4} - 1$

Step 4

The result can be shown in multiple forms.

Exact Form:

$e^{4} - 1$

Decimal Form:

$53.59815003 \dots$

Step 5