Calculus Examples

$\int_{0}^{1} 7 x e^{5 x} d x$

Step 1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int_{0}^{1} x e^{5 x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{5 x}$ .

$7 (x (\frac{1}{5} e^{5 x})]_{0}^{1} - \int_{0}^{1} \frac{1}{5} e^{5 x} d x)$

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{5}$ and $e^{5 x}$ .

$7 (x \frac{e^{5 x}}{5}]_{0}^{1} - \int_{0}^{1} \frac{1}{5} e^{5 x} d x)$

Step 3.2

Combine $x$ and $\frac{e^{5 x}}{5}$ .

$7 (\frac{x e^{5 x}}{5}]_{0}^{1} - \int_{0}^{1} \frac{1}{5} e^{5 x} d x)$

Step 3.3

Combine $\frac{1}{5}$ and $e^{5 x}$ .

$7 (\frac{x e^{5 x}}{5}]_{0}^{1} - \int_{0}^{1} \frac{e^{5 x}}{5} d x)$

Step 4

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$7 (\frac{x e^{5 x}}{5}]_{0}^{1} - (\frac{1}{5} \int_{0}^{1} e^{5 x} d x))$

Step 5

Let $u = 5 x$ . Then $d u = 5 d x$ , so $\frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 5 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 5.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 5.1.3

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

$5 \cdot 1$

Step 5.1.4

Multiply $5$ by $1$ .

$5$

Step 5.2

Substitute the lower limit in for $x$ in $u = 5 x$ .

$u_{lower} = 5 \cdot 0$

Step 5.3

Multiply $5$ by $0$ .

$u_{lower} = 0$

Step 5.4

Substitute the upper limit in for $x$ in $u = 5 x$ .

$u_{upper} = 5 \cdot 1$

Step 5.5

Multiply $5$ by $1$ .

$u_{upper} = 5$

Step 5.6

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

$u_{lower} = 0$

$u_{upper} = 5$

Step 5.7

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

$7 (\frac{x e^{5 x}}{5}]_{0}^{1} - \frac{1}{5} \int_{0}^{5} e^{u} \frac{1}{5} d u)$

Step 6

Combine $e^{u}$ and $\frac{1}{5}$ .

$7 (\frac{x e^{5 x}}{5}]_{0}^{1} - \frac{1}{5} \int_{0}^{5} \frac{e^{u}}{5} d u)$

Step 7

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$7 (\frac{x e^{5 x}}{5}]_{0}^{1} - \frac{1}{5} (\frac{1}{5} \int_{0}^{5} e^{u} d u))$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$7 (\frac{x e^{5 x}}{5}]_{0}^{1} - \frac{1}{5 \cdot 5} \int_{0}^{5} e^{u} d u)$

Step 8.2

Multiply $5$ by $5$ .

$7 (\frac{x e^{5 x}}{5}]_{0}^{1} - \frac{1}{25} \int_{0}^{5} e^{u} d u)$

Step 9

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

$7 (\frac{x e^{5 x}}{5}]_{0}^{1} - \frac{1}{25} e^{u}]_{0}^{5})$

Step 10

Combine $e^{u}]_{0}^{5}$ and $\frac{1}{25}$ .

$7 (\frac{x e^{5 x}}{5}]_{0}^{1} - \frac{e^{u}]_{0}^{5}}{25})$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $\frac{x e^{5 x}}{5}$ at $1$ and at $0$ .

$7 ((\frac{1 e^{5 \cdot 1}}{5}) - \frac{0 e^{5 \cdot 0}}{5} - \frac{e^{u}]_{0}^{5}}{25})$

Step 11.2

Evaluate $e^{u}$ at $5$ and at $0$ .

$7 ((\frac{1 e^{5 \cdot 1}}{5}) - \frac{0 e^{5 \cdot 0}}{5} - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3

Simplify.

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Step 11.3.1

Multiply $5$ by $1$ .

$7 (\frac{1 e^{5}}{5} - \frac{0 e^{5 \cdot 0}}{5} - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.2

Multiply $e^{5}$ by $1$ .

$7 (\frac{e^{5}}{5} - \frac{0 e^{5 \cdot 0}}{5} - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.3

Multiply $5$ by $0$ .

$7 (\frac{e^{5}}{5} - \frac{0 e^{0}}{5} - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.4

Anything raised to $0$ is $1$ .

$7 (\frac{e^{5}}{5} - \frac{0 \cdot 1}{5} - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.5

Multiply $0$ by $1$ .

$7 (\frac{e^{5}}{5} - \frac{0}{5} - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.6

Cancel the common factor of $0$ and $5$ .

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Step 11.3.6.1

Factor $5$ out of $0$ .

$7 (\frac{e^{5}}{5} - \frac{5 (0)}{5} - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.6.2

Cancel the common factors.

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Step 11.3.6.2.1

Factor $5$ out of $5$ .

$7 (\frac{e^{5}}{5} - \frac{5 \cdot 0}{5 \cdot 1} - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.6.2.2

Cancel the common factor.

$7 (\frac{e^{5}}{5} - \frac{5 \cdot 0}{5 \cdot 1} - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.6.2.3

Rewrite the expression.

$7 (\frac{e^{5}}{5} - \frac{0}{1} - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.6.2.4

Divide $0$ by $1$ .

$7 (\frac{e^{5}}{5} - 0 - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.7

Multiply $- 1$ by $0$ .

$7 (\frac{e^{5}}{5} + 0 - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.8

Add $\frac{e^{5}}{5}$ and $0$ .

$7 (\frac{e^{5}}{5} - \frac{(e^{5}) - e^{0}}{25})$

Step 11.3.9

Anything raised to $0$ is $1$ .

$7 (\frac{e^{5}}{5} - \frac{e^{5} - 1 \cdot 1}{25})$

Step 11.3.10

Multiply $- 1$ by $1$ .

$7 (\frac{e^{5}}{5} - \frac{e^{5} - 1}{25})$

Step 11.3.11

To write $\frac{e^{5}}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$7 (\frac{e^{5}}{5} \cdot \frac{5}{5} - \frac{e^{5} - 1}{25})$

Step 11.3.12

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.3.12.1

Multiply $\frac{e^{5}}{5}$ by $\frac{5}{5}$ .

$7 (\frac{e^{5} \cdot 5}{5 \cdot 5} - \frac{e^{5} - 1}{25})$

Step 11.3.12.2

Multiply $5$ by $5$ .

$7 (\frac{e^{5} \cdot 5}{25} - \frac{e^{5} - 1}{25})$

Step 11.3.13

Combine the numerators over the common denominator.

$7 \frac{e^{5} \cdot 5 - (e^{5} - 1)}{25}$

Step 11.3.14

Move $5$ to the left of $e^{5}$ .

$7 \frac{5 \cdot e^{5} - (e^{5} - 1)}{25}$

Step 11.3.15

Combine $7$ and $\frac{5 e^{5} - (e^{5} - 1)}{25}$ .

$\frac{7 (5 e^{5} - (e^{5} - 1))}{25}$

Step 12

Simplify the numerator.

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Step 12.1

Apply the distributive property.

$\frac{7 (5 e^{5} - e^{5} - - 1)}{25}$

Step 12.2

Multiply $- 1$ by $- 1$ .

$\frac{7 (5 e^{5} - e^{5} + 1)}{25}$

Step 12.3

Subtract $e^{5}$ from $5 e^{5}$ .

$\frac{7 (4 e^{5} + 1)}{25}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{7 (4 e^{5} + 1)}{25}$

Decimal Form:

$166.50273819 \dots$

Step 14