Calculus Examples

$\int_{0}^{1} 8 e^{2 x} d x$

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int_{0}^{1} e^{2 x} d x$

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $2 x$ .

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

Step 2.1.2

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 2.1.3

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 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

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

$u_{lower} = 2 \cdot 0$

Step 2.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 2.4

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

$u_{upper} = 2 \cdot 1$

Step 2.5

Multiply $2$ by $1$ .

$u_{upper} = 2$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = 2$

Step 2.7

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

$8 \int_{0}^{2} e^{u} \frac{1}{2} d u$

Step 3

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

$8 \int_{0}^{2} \frac{e^{u}}{2} d u$

Step 4

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

$8 (\frac{1}{2} \int_{0}^{2} e^{u} d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{2}$ and $8$ .

$\frac{8}{2} \int_{0}^{2} e^{u} d u$

Step 5.2

Cancel the common factor of $8$ and $2$ .

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Step 5.2.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 4}{2} \int_{0}^{2} e^{u} d u$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 4}{2 (1)} \int_{0}^{2} e^{u} d u$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot 1} \int_{0}^{2} e^{u} d u$

Step 5.2.2.3

Rewrite the expression.

$\frac{4}{1} \int_{0}^{2} e^{u} d u$

Step 5.2.2.4

Divide $4$ by $1$ .

$4 \int_{0}^{2} e^{u} d u$

Step 6

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

$4 (e^{u}]_{0}^{2})$

Step 7

Substitute and simplify.

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Step 7.1

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

$4 ((e^{2}) - e^{0})$

Step 7.2

Simplify.

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Step 7.2.1

Anything raised to $0$ is $1$ .

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

Step 7.2.2

Multiply $- 1$ by $1$ .

$4 (e^{2} - 1)$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$4 e^{2} + 4 \cdot - 1$

Step 8.2

Multiply $4$ by $- 1$ .

$4 e^{2} - 4$

Step 9

The result can be shown in multiple forms.

Exact Form:

$4 e^{2} - 4$

Decimal Form:

$25.55622439 \dots$

Step 10