Calculus Examples

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

Step 1

Split the integral at $0$ and write as a sum of limits.

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

Step 2

$\int_{t}^{0} e^{- x^{2}} d x$ is a special integral. The integral is the error function.

$lim_{t \to - \infty} erf (x)]_{t}^{0} + lim_{t \to \infty} \int_{0}^{t} e^{- x^{2}} d x$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $erf (x)$ at $0$ and at $t$ .

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

Step 3.2

Subtract $erf (x)$ from $erf (x)$ .

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

Step 4

$\int_{0}^{t} e^{- x^{2}} d x$ is a special integral. The integral is the error function.

$lim_{t \to - \infty} 0 + lim_{t \to \infty} erf (x)]_{0}^{t}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $erf (x)$ at $t$ and at $0$ .

$lim_{t \to - \infty} 0 + lim_{t \to \infty} (erf (x)) - erf (x)$

Step 5.2

Subtract $erf (x)$ from $erf (x)$ .

$lim_{t \to - \infty} 0 + lim_{t \to \infty} 0$

Step 6

Evaluate the limits.

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Step 6.1

Evaluate the limit of $0$ which is constant as $t$ approaches $- \infty$ .

$0 + lim_{t \to \infty} 0$

Step 6.2

Evaluate the limit of $0$ which is constant as $t$ approaches $\infty$ .

$0 + 0$

Step 6.3

Add $0$ and $0$ .

$0$