Calculus Examples

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

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

Step 1

Write the integral as a limit as

t

$t$ approaches

\infty

$\infty$ .

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

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

Step 2

The integral of

e^{x}

$e^{x}$ with respect to

x

$x$ is

e^{x}

$e^{x}$ .

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

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

Step 3

Substitute and simplify.

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Step 3.1

Evaluate

e^{x}

$e^{x}$ at

t

$t$ and at

1

$1$ .

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

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

Step 3.2

Simplify.

lim t \to \infty e^{t} - e

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

lim t \to \infty e^{t} - e

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

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

$t$ approaches

\infty

$\infty$ .

lim t \to \infty e^{t} - lim t \to \infty e

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

Step 4.2

Since the exponent

t

$t$ approaches

\infty

$\infty$ , the quantity

e^{t}

$e^{t}$ approaches

\infty

$\infty$ .

\infty - lim t \to \infty e

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

Step 4.3

Evaluate the limit of

e

$e$ which is constant as

t

$t$ approaches

\infty

$\infty$ .

\infty - e

$\infty - e$

Step 4.4

Infinity plus or minus a number is infinity.

\infty

$\infty$

\infty

$\infty$

⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$