Calculus Examples

$\int_{0}^{\infty} e^{- 2 x} d x$

Step 1

Write the integral as a limit as $t$ approaches $\infty$ .

$lim_{t \to \infty} \int_{0}^{t} e^{- 2 x} d x$

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 $u = - 2 x$ .

$u_{upper} = - 2 t$

Step 2.5

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

$u_{lower} = 0$

$u_{upper} = - 2 t$

Step 2.6

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

$lim_{t \to \infty} \int_{0}^{- 2 t} e^{u} \frac{1}{- 2} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$lim_{t \to \infty} \int_{0}^{- 2 t} e^{u} (- \frac{1}{2}) d u$

Step 3.2

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

$lim_{t \to \infty} \int_{0}^{- 2 t} - \frac{e^{u}}{2} d u$

Step 4

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

$lim_{t \to \infty} - \int_{0}^{- 2 t} \frac{e^{u}}{2} d u$

Step 5

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

$lim_{t \to \infty} - (\frac{1}{2} \int_{0}^{- 2 t} e^{u} d u)$

Step 6

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

$lim_{t \to \infty} - \frac{1}{2} e^{u}]_{0}^{- 2 t}$

Step 7

Combine $e^{u}]_{0}^{- 2 t}$ and $\frac{1}{2}$ .

$lim_{t \to \infty} - \frac{e^{u}]_{0}^{- 2 t}}{2}$

Step 8

Substitute and simplify.

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Step 8.1

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

$lim_{t \to \infty} - \frac{(e^{- 2 t}) - e^{0}}{2}$

Step 8.2

Simplify.

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Step 8.2.1

Anything raised to $0$ is $1$ .

$lim_{t \to \infty} - \frac{e^{- 2 t} - 1 \cdot 1}{2}$

Step 8.2.2

Multiply $- 1$ by $1$ .

$lim_{t \to \infty} - \frac{e^{- 2 t} - 1}{2}$

Step 9

Evaluate the limit.

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Step 9.1

Evaluate the limit.

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Step 9.1.1

Move the term $- 1$ outside of the limit because it is constant with respect to $t$ .

$- lim_{t \to \infty} \frac{e^{- 2 t} - 1}{2}$

Step 9.1.2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $t$ .

$- \frac{1}{2} lim_{t \to \infty} e^{- 2 t} - 1$

Step 9.1.3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $\infty$ .

$- \frac{1}{2} (lim_{t \to \infty} e^{- 2 t} - lim_{t \to \infty} 1)$

Step 9.2

Since the exponent $- 2 t$ approaches $- \infty$ , the quantity $e^{- 2 t}$ approaches $0$ .

$- \frac{1}{2} (0 - lim_{t \to \infty} 1)$

Step 9.3

Evaluate the limit.

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Step 9.3.1

Evaluate the limit of $1$ which is constant as $t$ approaches $\infty$ .

$- \frac{1}{2} (0 - 1 \cdot 1)$

Step 9.3.2

Simplify the answer.

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Step 9.3.2.1

Multiply $- 1$ by $1$ .

$- \frac{1}{2} (0 - 1)$

Step 9.3.2.2

Subtract $1$ from $0$ .

$- \frac{1}{2} \cdot - 1$

Step 9.3.2.3

Multiply $- \frac{1}{2} \cdot - 1$ .

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Step 9.3.2.3.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2})$

Step 9.3.2.3.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$