Calculus Examples

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

Step 1

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

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

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 $u = 2 x$ .

$u_{lower} = 2 t$

Step 2.3

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

$u_{upper} = 2 \cdot 0$

Step 2.4

Multiply $2$ by $0$ .

$u_{upper} = 0$

Step 2.5

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

$u_{lower} = 2 t$

$u_{upper} = 0$

Step 2.6

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Substitute and simplify.

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Step 7.1

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

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

Step 7.2

Anything raised to $0$ is $1$ .

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

Step 8

Evaluate the limit.

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Step 8.1

Evaluate the limit.

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Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.1.3

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

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

Step 8.2

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

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

Step 8.3

Simplify the answer.

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Step 8.3.1

Subtract $0$ from $1$ .

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

Step 8.3.2

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

$\frac{1}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$