Calculus Examples

$\int_{0}^{1} 5 x e^{2 x} d x$

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{0}^{1} x e^{2 x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{2 x}$ .

$5 (x (\frac{1}{2} e^{2 x})]_{0}^{1} - \int_{0}^{1} \frac{1}{2} e^{2 x} d x)$

Step 3

Simplify.

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Step 3.1

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

$5 (x \frac{e^{2 x}}{2}]_{0}^{1} - \int_{0}^{1} \frac{1}{2} e^{2 x} d x)$

Step 3.2

Combine $x$ and $\frac{e^{2 x}}{2}$ .

$5 (\frac{x e^{2 x}}{2}]_{0}^{1} - \int_{0}^{1} \frac{1}{2} e^{2 x} d x)$

Step 3.3

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

$5 (\frac{x e^{2 x}}{2}]_{0}^{1} - \int_{0}^{1} \frac{e^{2 x}}{2} d x)$

Step 4

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

$5 (\frac{x e^{2 x}}{2}]_{0}^{1} - (\frac{1}{2} \int_{0}^{1} e^{2 x} d x))$

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $2 x$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

$2 \cdot 1$

Step 5.1.4

Multiply $2$ by $1$ .

$2$

Step 5.2

Substitute the lower limit in for $x$ in $u = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 5.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 5.4

Substitute the upper limit in for $x$ in $u = 2 x$ .

$u_{upper} = 2 \cdot 1$

Step 5.5

Multiply $2$ by $1$ .

$u_{upper} = 2$

Step 5.6

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

$u_{lower} = 0$

$u_{upper} = 2$

Step 5.7

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

$5 (\frac{x e^{2 x}}{2}]_{0}^{1} - \frac{1}{2} \int_{0}^{2} e^{u} \frac{1}{2} d u)$

Step 6

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

$5 (\frac{x e^{2 x}}{2}]_{0}^{1} - \frac{1}{2} \int_{0}^{2} \frac{e^{u}}{2} d u)$

Step 7

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

$5 (\frac{x e^{2 x}}{2}]_{0}^{1} - \frac{1}{2} (\frac{1}{2} \int_{0}^{2} e^{u} d u))$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$5 (\frac{x e^{2 x}}{2}]_{0}^{1} - \frac{1}{2 \cdot 2} \int_{0}^{2} e^{u} d u)$

Step 8.2

Multiply $2$ by $2$ .

$5 (\frac{x e^{2 x}}{2}]_{0}^{1} - \frac{1}{4} \int_{0}^{2} e^{u} d u)$

Step 9

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

$5 (\frac{x e^{2 x}}{2}]_{0}^{1} - \frac{1}{4} e^{u}]_{0}^{2})$

Step 10

Combine $e^{u}]_{0}^{2}$ and $\frac{1}{4}$ .

$5 (\frac{x e^{2 x}}{2}]_{0}^{1} - \frac{e^{u}]_{0}^{2}}{4})$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $\frac{x e^{2 x}}{2}$ at $1$ and at $0$ .

$5 ((\frac{1 e^{2 \cdot 1}}{2}) - \frac{0 e^{2 \cdot 0}}{2} - \frac{e^{u}]_{0}^{2}}{4})$

Step 11.2

Evaluate $e^{u}$ at $2$ and at $0$ .

$5 ((\frac{1 e^{2 \cdot 1}}{2}) - \frac{0 e^{2 \cdot 0}}{2} - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3

Simplify.

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Step 11.3.1

Multiply $2$ by $1$ .

$5 (\frac{1 e^{2}}{2} - \frac{0 e^{2 \cdot 0}}{2} - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.2

Multiply $e^{2}$ by $1$ .

$5 (\frac{e^{2}}{2} - \frac{0 e^{2 \cdot 0}}{2} - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.3

Multiply $2$ by $0$ .

$5 (\frac{e^{2}}{2} - \frac{0 e^{0}}{2} - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.4

Anything raised to $0$ is $1$ .

$5 (\frac{e^{2}}{2} - \frac{0 \cdot 1}{2} - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.5

Multiply $0$ by $1$ .

$5 (\frac{e^{2}}{2} - \frac{0}{2} - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.6

Cancel the common factor of $0$ and $2$ .

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Step 11.3.6.1

Factor $2$ out of $0$ .

$5 (\frac{e^{2}}{2} - \frac{2 (0)}{2} - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.6.2

Cancel the common factors.

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Step 11.3.6.2.1

Factor $2$ out of $2$ .

$5 (\frac{e^{2}}{2} - \frac{2 \cdot 0}{2 \cdot 1} - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.6.2.2

Cancel the common factor.

$5 (\frac{e^{2}}{2} - \frac{2 \cdot 0}{2 \cdot 1} - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.6.2.3

Rewrite the expression.

$5 (\frac{e^{2}}{2} - \frac{0}{1} - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.6.2.4

Divide $0$ by $1$ .

$5 (\frac{e^{2}}{2} - 0 - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.7

Multiply $- 1$ by $0$ .

$5 (\frac{e^{2}}{2} + 0 - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.8

Add $\frac{e^{2}}{2}$ and $0$ .

$5 (\frac{e^{2}}{2} - \frac{(e^{2}) - e^{0}}{4})$

Step 11.3.9

Anything raised to $0$ is $1$ .

$5 (\frac{e^{2}}{2} - \frac{e^{2} - 1 \cdot 1}{4})$

Step 11.3.10

Multiply $- 1$ by $1$ .

$5 (\frac{e^{2}}{2} - \frac{e^{2} - 1}{4})$

Step 11.3.11

To write $\frac{e^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 (\frac{e^{2}}{2} \cdot \frac{2}{2} - \frac{e^{2} - 1}{4})$

Step 11.3.12

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.3.12.1

Multiply $\frac{e^{2}}{2}$ by $\frac{2}{2}$ .

$5 (\frac{e^{2} \cdot 2}{2 \cdot 2} - \frac{e^{2} - 1}{4})$

Step 11.3.12.2

Multiply $2$ by $2$ .

$5 (\frac{e^{2} \cdot 2}{4} - \frac{e^{2} - 1}{4})$

Step 11.3.13

Combine the numerators over the common denominator.

$5 \frac{e^{2} \cdot 2 - (e^{2} - 1)}{4}$

Step 11.3.14

Move $2$ to the left of $e^{2}$ .

$5 \frac{2 \cdot e^{2} - (e^{2} - 1)}{4}$

Step 11.3.15

Combine $5$ and $\frac{2 e^{2} - (e^{2} - 1)}{4}$ .

$\frac{5 (2 e^{2} - (e^{2} - 1))}{4}$

Step 12

Simplify the numerator.

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Step 12.1

Apply the distributive property.

$\frac{5 (2 e^{2} - e^{2} - - 1)}{4}$

Step 12.2

Multiply $- 1$ by $- 1$ .

$\frac{5 (2 e^{2} - e^{2} + 1)}{4}$

Step 12.3

Subtract $e^{2}$ from $2 e^{2}$ .

$\frac{5 (e^{2} + 1)}{4}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{5 (e^{2} + 1)}{4}$

Decimal Form:

$10.48632012 \dots$

Step 14