Calculus Examples

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

Step 1

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

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

Step 2

Let $u_{2} = e^{- x^{2}}$ . Then $d u_{2} = - 2 x e^{- x^{2}} d x$ , so $- \frac{1}{2} d u_{2} = x e^{- x^{2}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.1

Let $u_{2} = e^{- x^{2}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.1.1

Differentiate $e^{- x^{2}}$ .

$\frac{d}{d x} [e^{- x^{2}}]$

Step 2.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - x^{2}$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $- x^{2}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- x^{2}]$

Step 2.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [- x^{2}]$

Step 2.1.2.3

Replace all occurrences of $u_{1}$ with $- x^{2}$ .

$e^{- x^{2}} \frac{d}{d x} [- x^{2}]$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$e^{- x^{2}} (- \frac{d}{d x} [x^{2}])$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$e^{- x^{2}} (- (2 x))$

Step 2.1.3.3

Multiply $2$ by $- 1$ .

$e^{- x^{2}} (- 2 x)$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Reorder the factors of $e^{- x^{2}} (- 2 x)$ .

$- 2 e^{- x^{2}} x$

Step 2.1.4.2

Reorder factors in $- 2 e^{- x^{2}} x$ .

$- 2 x e^{- x^{2}}$

Step 2.2

Substitute the lower limit in for $x$ in $u_{2} = e^{- x^{2}}$ .

$u_{lower} = e^{- t^{2}}$

Step 2.3

Substitute the upper limit in for $x$ in $u_{2} = e^{- x^{2}}$ .

$u_{upper} = e^{- 0^{2}}$

Step 2.4

Simplify.

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Step 2.4.1

Raising $0$ to any positive power yields $0$ .

$u_{upper} = e^{- 0}$

Step 2.4.2

Multiply $- 1$ by $0$ .

$u_{upper} = e^{0}$

Step 2.4.3

Anything raised to $0$ is $1$ .

$u_{upper} = 1$

Step 2.5

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

$u_{lower} = e^{- t^{2}}$

$u_{upper} = 1$

Step 2.6

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

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

Step 3

Move the negative in front of the fraction.

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

Step 4

Apply the constant rule.

$lim_{t \to - \infty} - \frac{1}{2} u_{2}]_{e^{- t^{2}}}^{1} + lim_{t \to \infty} \int_{0}^{t} x e^{- x^{2}} d x$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $- \frac{1}{2} u_{2}$ at $1$ and at $e^{- t^{2}}$ .

$lim_{t \to - \infty} (- \frac{1}{2} \cdot 1) + \frac{1}{2} e^{- t^{2}} + lim_{t \to \infty} \int_{0}^{t} x e^{- x^{2}} d x$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $- 1$ by $1$ .

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

Step 5.2.2

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

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

Step 6

Let $u_{2} = e^{- x^{2}}$ . Then $d u_{2} = - 2 x e^{- x^{2}} d x$ , so $- \frac{1}{2} d u_{2} = x e^{- x^{2}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.1

Let $u_{2} = e^{- x^{2}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 6.1.1

Differentiate $e^{- x^{2}}$ .

$\frac{d}{d x} [e^{- x^{2}}]$

Step 6.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - x^{2}$ .

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Step 6.1.2.1

To apply the Chain Rule, set $u_{1}$ as $- x^{2}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- x^{2}]$

Step 6.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [- x^{2}]$

Step 6.1.2.3

Replace all occurrences of $u_{1}$ with $- x^{2}$ .

$e^{- x^{2}} \frac{d}{d x} [- x^{2}]$

Step 6.1.3

Differentiate.

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Step 6.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$e^{- x^{2}} (- \frac{d}{d x} [x^{2}])$

Step 6.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$e^{- x^{2}} (- (2 x))$

Step 6.1.3.3

Multiply $2$ by $- 1$ .

$e^{- x^{2}} (- 2 x)$

Step 6.1.4

Simplify.

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Step 6.1.4.1

Reorder the factors of $e^{- x^{2}} (- 2 x)$ .

$- 2 e^{- x^{2}} x$

Step 6.1.4.2

Reorder factors in $- 2 e^{- x^{2}} x$ .

$- 2 x e^{- x^{2}}$

Step 6.2

Substitute the lower limit in for $x$ in $u_{2} = e^{- x^{2}}$ .

$u_{lower} = e^{- 0^{2}}$

Step 6.3

Simplify.

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Step 6.3.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = e^{- 0}$

Step 6.3.2

Multiply $- 1$ by $0$ .

$u_{lower} = e^{0}$

Step 6.3.3

Anything raised to $0$ is $1$ .

$u_{lower} = 1$

Step 6.4

Substitute the upper limit in for $x$ in $u_{2} = e^{- x^{2}}$ .

$u_{upper} = e^{- t^{2}}$

Step 6.5

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

$u_{lower} = 1$

$u_{upper} = e^{- t^{2}}$

Step 6.6

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Apply the constant rule.

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

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $- \frac{1}{2} u_{2}$ at $e^{- t^{2}}$ and at $1$ .

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

Step 9.2

Simplify.

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 10

Evaluate the limits.

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Step 10.1

Combine fractions using a common denominator.

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Step 10.1.1

Combine the numerators over the common denominator.

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

Step 10.1.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 10.1.3

Factor $- 1$ out of $e^{- t^{2}}$ .

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

Step 10.1.4

Factor $- 1$ out of $- 1 (1) - 1 (- e^{- t^{2}})$ .

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

Step 10.1.5

Move the negative in front of the fraction.

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

Step 10.2

Combine fractions using a common denominator.

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Step 10.2.1

Combine the numerators over the common denominator.

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

Step 10.2.2

Factor $- 1$ out of $- e^{- t^{2}}$ .

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

Step 10.2.3

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 10.2.4

Factor $- 1$ out of $- (e^{- t^{2}}) - 1 (- 1)$ .

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

Step 10.2.5

Rewrite $- (e^{- t^{2}} - 1)$ as $- 1 (e^{- t^{2}} - 1)$ .

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

Step 10.2.6

Move the negative in front of the fraction.

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

Step 10.3

Evaluate the limit.

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Step 10.3.1

Move the term $- 1$ outside of the limit because it is constant with respect to $t$ .

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

Step 10.3.2

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

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

Step 10.3.3

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

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

Step 10.3.4

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

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

Step 10.4

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

$- \frac{1}{2} (1 - 0) + lim_{t \to \infty} - \frac{e^{- t^{2}} - 1}{2}$

Step 10.5

Evaluate the limit.

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Step 10.5.1

Move the term $- 1$ outside of the limit because it is constant with respect to $t$ .

$- \frac{1}{2} (1 - 0) - lim_{t \to \infty} \frac{e^{- t^{2}} - 1}{2}$

Step 10.5.2

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

$- \frac{1}{2} (1 - 0) - \frac{1}{2} lim_{t \to \infty} e^{- t^{2}} - 1$

Step 10.5.3

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

$- \frac{1}{2} (1 - 0) - \frac{1}{2} (lim_{t \to \infty} e^{- t^{2}} - lim_{t \to \infty} 1)$

Step 10.6

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

$- \frac{1}{2} (1 - 0) - \frac{1}{2} (0 - lim_{t \to \infty} 1)$

Step 10.7

Evaluate the limit.

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Step 10.7.1

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

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

Step 10.7.2

Simplify the answer.

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Step 10.7.2.1

Simplify each term.

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Step 10.7.2.1.1

Subtract $0$ from $1$ .

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

Step 10.7.2.1.2

Multiply $- 1$ by $1$ .

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

Step 10.7.2.1.3

Multiply $- 1$ by $1$ .

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

Step 10.7.2.1.4

Subtract $1$ from $0$ .

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

Step 10.7.2.1.5

Multiply $- \frac{1}{2} \cdot - 1$ .

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Step 10.7.2.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 10.7.2.1.5.2

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

$- \frac{1}{2} + \frac{1}{2}$

Step 10.7.2.2

Combine the numerators over the common denominator.

$\frac{- 1 + 1}{2}$

Step 10.7.2.3

Add $- 1$ and $1$ .

$\frac{0}{2}$

Step 10.7.2.4

Divide $0$ by $2$ .

$0$