Calculus Examples

$\int_{0}^{1} 2 e^{0.09 x} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{0}^{1} e^{0.09 x} d x$

Step 2

Let $u = 0.09 x$ . Then $d u = 0.09 d x$ , so $\frac{1}{0.09} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 0.09 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $0.09 x$ .

$\frac{d}{d x} [0.09 x]$

Step 2.1.2

Since $0.09$ is constant with respect to $x$ , the derivative of $0.09 x$ with respect to $x$ is $0.09 \frac{d}{d x} [x]$ .

$0.09 \frac{d}{d x} [x]$

Step 2.1.3

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

$0.09 \cdot 1$

Step 2.1.4

Multiply $0.09$ by $1$ .

$0.09$

Step 2.2

Substitute the lower limit in for $x$ in $u = 0.09 x$ .

$u_{lower} = 0.09 \cdot 0$

Step 2.3

Multiply $0.09$ by $0$ .

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u = 0.09 x$ .

$u_{upper} = 0.09 \cdot 1$

Step 2.5

Multiply $0.09$ by $1$ .

$u_{upper} = 0.09$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = 0.09$

Step 2.7

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

$2 \int_{0}^{0.09} e^{u} \frac{1}{0.09} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{0.09}$ .

$2 \int_{0}^{0.09} \frac{e^{u}}{0.09} d u$

Step 4

Since $\frac{1}{0.09}$ is constant with respect to $u$ , move $\frac{1}{0.09}$ out of the integral.

$2 (\frac{1}{0.09} \int_{0}^{0.09} e^{u} d u)$

Step 5

Combine $\frac{1}{0.09}$ and $2$ .

$\frac{2}{0.09} \int_{0}^{0.09} e^{u} d u$

Step 6

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

$\frac{2}{0.09} e^{u}]_{0}^{0.09}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $e^{u}$ at $0.09$ and at $0$ .

$\frac{2}{0.09} ((e^{0.09}) - e^{0})$

Step 7.2

Simplify.

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Step 7.2.1

Anything raised to $0$ is $1$ .

$\frac{2}{0.09} (e^{0.09} - 1 \cdot 1)$

Step 7.2.2

Multiply $- 1$ by $1$ .

$\frac{2}{0.09} (e^{0.09} - 1)$

Step 8

Simplify.

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Step 8.1

Divide $2$ by $0.09$ .

$22.\overline{2} (e^{0.09} - 1)$

Step 8.2

Multiply $22.\overline{2}$ by $e^{0.09} - 1$ .

$2.09276186$

Step 9