Calculus Examples

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

Step 1

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

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

Step 2

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

$lim_{t \to - \infty} e^{x}]_{t}^{0}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $e^{x}$ at $0$ and at $t$ .

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

Step 3.2

Anything raised to $0$ is $1$ .

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

Step 4

Evaluate the limit.

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Step 4.1

Evaluate the limit.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

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

$1 - 0$

Step 4.3

Subtract $0$ from $1$ .

$1$