Calculus Examples

$\int_{1}^{\infty} e^{x} d x$

Step 1

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

$lim_{t \to \infty} \int_{1}^{t} e^{x} d x$

Step 2

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

$lim_{t \to \infty} e^{x}]_{1}^{t}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $e^{x}$ at $t$ and at $1$ .

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

Step 3.2

Simplify.

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

$\infty - e$

Step 4.4

Infinity plus or minus a number is infinity.

$\infty$