Calculus Examples

$\int_{0}^{1} x^{4} e^{2 x^{5} - 7} d x$

Step 1

Let $u = 2 x^{5} - 7$ . Then $d u = 10 x^{4} d x$ , so $\frac{1}{10} d u = x^{4} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x^{5} - 7$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x^{5} - 7$ .

$\frac{d}{d x} [2 x^{5} - 7]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x^{5} - 7$ with respect to $x$ is $\frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [- 7]$ .

$\frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [- 7]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x^{5}]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{5}$ with respect to $x$ is $2 \frac{d}{d x} [x^{5}]$ .

$2 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 7]$

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 $n = 5$ .

$2 (5 x^{4}) + \frac{d}{d x} [- 7]$

Step 1.1.3.3

Multiply $5$ by $2$ .

$10 x^{4} + \frac{d}{d x} [- 7]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$10 x^{4} + 0$

Step 1.1.4.2

Add $10 x^{4}$ and $0$ .

$10 x^{4}$

Step 1.2

Substitute the lower limit in for $x$ in $u = 2 x^{5} - 7$ .

$u_{lower} = 2 \cdot 0^{5} - 7$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 2 \cdot 0 - 7$

Step 1.3.1.2

Multiply $2$ by $0$ .

$u_{lower} = 0 - 7$

Step 1.3.2

Subtract $7$ from $0$ .

$u_{lower} = - 7$

Step 1.4

Substitute the upper limit in for $x$ in $u = 2 x^{5} - 7$ .

$u_{upper} = 2 \cdot 1^{5} - 7$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$u_{upper} = 2 \cdot 1 - 7$

Step 1.5.1.2

Multiply $2$ by $1$ .

$u_{upper} = 2 - 7$

Step 1.5.2

Subtract $7$ from $2$ .

$u_{upper} = - 5$

Step 1.6

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

$u_{lower} = - 7$

$u_{upper} = - 5$

Step 1.7

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

$\int_{- 7}^{- 5} e^{u} \frac{1}{10} d u$

Step 2

Combine $e^{u}$ and $\frac{1}{10}$ .

$\int_{- 7}^{- 5} \frac{e^{u}}{10} d u$

Step 3

Since $\frac{1}{10}$ is constant with respect to $u$ , move $\frac{1}{10}$ out of the integral.

$\frac{1}{10} \int_{- 7}^{- 5} e^{u} d u$

Step 4

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

$\frac{1}{10} e^{u}]_{- 7}^{- 5}$

Step 5

Evaluate $e^{u}$ at $- 5$ and at $- 7$ .

$\frac{1}{10} (e^{- 5} - e^{- 7})$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{1}{10} e^{- 5} + \frac{1}{10} (- e^{- 7})$

Step 6.2

Combine $\frac{1}{10}$ and $e^{- 5}$ .

$\frac{e^{- 5}}{10} + \frac{1}{10} (- e^{- 7})$

Step 6.3

Combine $\frac{1}{10}$ and $e^{- 7}$ .

$\frac{e^{- 5}}{10} - \frac{e^{- 7}}{10}$

Step 6.4

Simplify each term.

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Step 6.4.1

Move $e^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{10 e^{5}} - \frac{e^{- 7}}{10}$

Step 6.4.2

Move $e^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{10 e^{5}} - \frac{1}{10 e^{7}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{10 e^{5}} - \frac{1}{10 e^{7}}$

Decimal Form:

$0.00058260 \dots$

Step 8