Calculus Examples

$\int_{1}^{3} (x + 1) e^{x^{2} + 2 x} d x$

Step 1

Let $u = x^{2} + 2 x$ . Then $d u = (2 x + 2) d x$ , so $\frac{1}{2} d u = (x + 1) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{2} + 2 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} + 2 x$ .

$\frac{d}{d x} [x^{2} + 2 x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{2} + 2 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [2 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 x + 2 \frac{d}{d x} [x]$

Step 1.1.3.2

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

$2 x + 2 \cdot 1$

Step 1.1.3.3

Multiply $2$ by $1$ .

$2 x + 2$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{2} + 2 x$ .

$u_{lower} = 1^{2} + 2 \cdot 1$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

One to any power is one.

$u_{lower} = 1 + 2 \cdot 1$

Step 1.3.1.2

Multiply $2$ by $1$ .

$u_{lower} = 1 + 2$

Step 1.3.2

Add $1$ and $2$ .

$u_{lower} = 3$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{2} + 2 x$ .

$u_{upper} = 3^{2} + 2 \cdot 3$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Raise $3$ to the power of $2$ .

$u_{upper} = 9 + 2 \cdot 3$

Step 1.5.1.2

Multiply $2$ by $3$ .

$u_{upper} = 9 + 6$

Step 1.5.2

Add $9$ and $6$ .

$u_{upper} = 15$

Step 1.6

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

$u_{lower} = 3$

$u_{upper} = 15$

Step 1.7

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

$\int_{3}^{15} e^{u} \frac{1}{2} d u$

Step 2

Combine $e^{u}$ and $\frac{1}{2}$ .

$\int_{3}^{15} \frac{e^{u}}{2} d u$

Step 3

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

$\frac{1}{2} \int_{3}^{15} e^{u} d u$

Step 4

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

$\frac{1}{2} e^{u}]_{3}^{15}$

Step 5

Evaluate $e^{u}$ at $15$ and at $3$ .

$\frac{1}{2} (e^{15} - e^{3})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2} \cdot (e^{15} - e^{3})$

Decimal Form:

$1634498.64346759$