Calculus Examples

$\int_{- 3}^{5} (5 x^{2} - e^{5 x}) d x$

Step 1

Remove parentheses.

$\int_{- 3}^{5} 5 x^{2} - e^{5 x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 3}^{5} 5 x^{2} d x + \int_{- 3}^{5} - e^{5 x} d x$

Step 3

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

$5 \int_{- 3}^{5} x^{2} d x + \int_{- 3}^{5} - e^{5 x} d x$

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$5 (\frac{1}{3} x^{3}]_{- 3}^{5}) + \int_{- 3}^{5} - e^{5 x} d x$

Step 5

Combine $\frac{1}{3}$ and $x^{3}$ .

$5 (\frac{x^{3}}{3}]_{- 3}^{5}) + \int_{- 3}^{5} - e^{5 x} d x$

Step 6

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

$5 (\frac{x^{3}}{3}]_{- 3}^{5}) - \int_{- 3}^{5} e^{5 x} d x$

Step 7

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 7.1

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

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Step 7.1.1

Differentiate $5 x$ .

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

Step 7.1.2

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 7.1.3

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 7.1.4

Multiply $5$ by $1$ .

$5$

Step 7.2

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

$u_{lower} = 5 \cdot - 3$

Step 7.3

Multiply $5$ by $- 3$ .

$u_{lower} = - 15$

Step 7.4

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

$u_{upper} = 5 \cdot 5$

Step 7.5

Multiply $5$ by $5$ .

$u_{upper} = 25$

Step 7.6

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

$u_{lower} = - 15$

$u_{upper} = 25$

Step 7.7

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

$5 (\frac{x^{3}}{3}]_{- 3}^{5}) - \int_{- 15}^{25} e^{u} \frac{1}{5} d u$

Step 8

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

$5 (\frac{x^{3}}{3}]_{- 3}^{5}) - \int_{- 15}^{25} \frac{e^{u}}{5} d u$

Step 9

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

$5 (\frac{x^{3}}{3}]_{- 3}^{5}) - (\frac{1}{5} \int_{- 15}^{25} e^{u} d u)$

Step 10

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

$5 (\frac{x^{3}}{3}]_{- 3}^{5}) - \frac{1}{5} e^{u}]_{- 15}^{25}$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $\frac{x^{3}}{3}$ at $5$ and at $- 3$ .

$5 ((\frac{5^{3}}{3}) - \frac{{(- 3)}^{3}}{3}) - \frac{1}{5} e^{u}]_{- 15}^{25}$

Step 11.2

Evaluate $e^{u}$ at $25$ and at $- 15$ .

$5 ((\frac{5^{3}}{3}) - \frac{{(- 3)}^{3}}{3}) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3

Simplify.

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Step 11.3.1

Raise $5$ to the power of $3$ .

$5 (\frac{125}{3} - \frac{{(- 3)}^{3}}{3}) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.2

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

$5 (\frac{125}{3} - \frac{- 27}{3}) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.3

Cancel the common factor of $- 27$ and $3$ .

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Step 11.3.3.1

Factor $3$ out of $- 27$ .

$5 (\frac{125}{3} - \frac{3 \cdot - 9}{3}) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.3.2

Cancel the common factors.

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Step 11.3.3.2.1

Factor $3$ out of $3$ .

$5 (\frac{125}{3} - \frac{3 \cdot - 9}{3 (1)}) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.3.2.2

Cancel the common factor.

$5 (\frac{125}{3} - \frac{3 \cdot - 9}{3 \cdot 1}) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.3.2.3

Rewrite the expression.

$5 (\frac{125}{3} - \frac{- 9}{1}) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.3.2.4

Divide $- 9$ by $1$ .

$5 (\frac{125}{3} - - 9) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.4

Multiply $- 1$ by $- 9$ .

$5 (\frac{125}{3} + 9) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.5

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$5 (\frac{125}{3} + 9 \cdot \frac{3}{3}) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.6

Combine $9$ and $\frac{3}{3}$ .

$5 (\frac{125}{3} + \frac{9 \cdot 3}{3}) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.7

Combine the numerators over the common denominator.

$5 \frac{125 + 9 \cdot 3}{3} - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.8

Simplify the numerator.

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Step 11.3.8.1

Multiply $9$ by $3$ .

$5 \frac{125 + 27}{3} - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.8.2

Add $125$ and $27$ .

$5 (\frac{152}{3}) - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.9

Combine $5$ and $\frac{152}{3}$ .

$\frac{5 \cdot 152}{3} - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.10

Multiply $5$ by $152$ .

$\frac{760}{3} - \frac{1}{5} ((e^{25}) - e^{- 15})$

Step 11.3.11

To write $- \frac{1}{5} (e^{25} - e^{- 15})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{760}{3} - \frac{1}{5} (e^{25} - e^{- 15}) \cdot \frac{3}{3}$

Step 11.3.12

Combine $- \frac{1}{5} (e^{25} - e^{- 15})$ and $\frac{3}{3}$ .

$\frac{760}{3} + \frac{- \frac{1}{5} (e^{25} - e^{- 15}) \cdot 3}{3}$

Step 11.3.13

Combine the numerators over the common denominator.

$\frac{760 - \frac{1}{5} (e^{25} - e^{- 15}) \cdot 3}{3}$

Step 11.3.14

Multiply $3$ by $- 1$ .

$\frac{760 - 3 (\frac{1}{5}) (e^{25} - e^{- 15})}{3}$

Step 11.3.15

Combine $- 3$ and $\frac{1}{5}$ .

$\frac{760 + \frac{- 3}{5} (e^{25} - e^{- 15})}{3}$

Step 11.3.16

Move the negative in front of the fraction.

$\frac{760 - \frac{3}{5} (e^{25} - e^{- 15})}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{760 - \frac{3}{5} \cdot (e^{25} - e^{- 15})}{3}$

Decimal Form:

$- 14400979614.1438$

Step 13