Calculus Examples

$\int_{0}^{\infty} x^{2} e^{- x^{3}} d x$

Step 1

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

$lim_{t \to \infty} \int_{0}^{t} x^{2} e^{- x^{3}} d x$

Step 2

Let $u_{2} = e^{- x^{3}}$ . Then $d u_{2} = - 3 x^{2} e^{- x^{3}} d x$ , so $- \frac{1}{3} d u_{2} = x^{2} e^{- x^{3}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.1

Let $u_{2} = e^{- x^{3}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.1.1

Differentiate $e^{- x^{3}}$ .

$\frac{d}{d x} [e^{- x^{3}}]$

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 $g (x) = - x^{3}$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $- x^{3}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- x^{3}]$

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 $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [- x^{3}]$

Step 2.1.2.3

Replace all occurrences of $u_{1}$ with $- x^{3}$ .

$e^{- x^{3}} \frac{d}{d x} [- x^{3}]$

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 $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$e^{- x^{3}} (- \frac{d}{d x} [x^{3}])$

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 $n = 3$ .

$e^{- x^{3}} (- (3 x^{2}))$

Step 2.1.3.3

Multiply $3$ by $- 1$ .

$e^{- x^{3}} (- 3 x^{2})$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Reorder the factors of $e^{- x^{3}} (- 3 x^{2})$ .

$- 3 e^{- x^{3}} x^{2}$

Step 2.1.4.2

Reorder factors in $- 3 e^{- x^{3}} x^{2}$ .

$- 3 x^{2} e^{- x^{3}}$

Step 2.2

Substitute the lower limit in for $x$ in $u_{2} = e^{- x^{3}}$ .

$u_{lower} = e^{- 0^{3}}$

Step 2.3

Simplify.

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Step 2.3.1

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

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

Step 2.3.2

Multiply $- 1$ by $0$ .

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

Step 2.3.3

Anything raised to $0$ is $1$ .

$u_{lower} = 1$

Step 2.4

Substitute the upper limit in for $x$ in $u_{2} = e^{- x^{3}}$ .

$u_{upper} = e^{- t^{3}}$

Step 2.5

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

$u_{lower} = 1$

$u_{upper} = e^{- t^{3}}$

Step 2.6

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

$lim_{t \to \infty} \int_{1}^{e^{- t^{3}}} \frac{1}{- 3} d u_{2}$

Step 3

Move the negative in front of the fraction.

$lim_{t \to \infty} \int_{1}^{e^{- t^{3}}} - \frac{1}{3} d u_{2}$

Step 4

Apply the constant rule.

$lim_{t \to \infty} - \frac{1}{3} u_{2}]_{1}^{e^{- t^{3}}}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $- \frac{1}{3} u_{2}$ at $e^{- t^{3}}$ and at $1$ .

$lim_{t \to \infty} (- \frac{1}{3} e^{- t^{3}}) + \frac{1}{3} \cdot 1$

Step 5.2

Simplify.

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Step 5.2.1

Combine $e^{- t^{3}}$ and $\frac{1}{3}$ .

$lim_{t \to \infty} - \frac{e^{- t^{3}}}{3} + \frac{1}{3} \cdot 1$

Step 5.2.2

Multiply $\frac{1}{3}$ by $1$ .

$lim_{t \to \infty} - \frac{e^{- t^{3}}}{3} + \frac{1}{3}$

Step 6

Evaluate the limit.

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Step 6.1

Combine fractions using a common denominator.

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Step 6.1.1

Combine the numerators over the common denominator.

$lim_{t \to \infty} \frac{- e^{- t^{3}} + 1}{3}$

Step 6.1.2

Factor $- 1$ out of $- e^{- t^{3}}$ .

$lim_{t \to \infty} \frac{- (e^{- t^{3}}) + 1}{3}$

Step 6.1.3

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

$lim_{t \to \infty} \frac{- (e^{- t^{3}}) - 1 (- 1)}{3}$

Step 6.1.4

Factor $- 1$ out of $- (e^{- t^{3}}) - 1 (- 1)$ .

$lim_{t \to \infty} \frac{- (e^{- t^{3}} - 1)}{3}$

Step 6.1.5

Rewrite $- (e^{- t^{3}} - 1)$ as $- 1 (e^{- t^{3}} - 1)$ .

$lim_{t \to \infty} \frac{- 1 (e^{- t^{3}} - 1)}{3}$

Step 6.1.6

Move the negative in front of the fraction.

$lim_{t \to \infty} - \frac{e^{- t^{3}} - 1}{3}$

Step 6.2

Evaluate the limit.

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Step 6.2.1

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

$- lim_{t \to \infty} \frac{e^{- t^{3}} - 1}{3}$

Step 6.2.2

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $t$ .

$- \frac{1}{3} lim_{t \to \infty} e^{- t^{3}} - 1$

Step 6.2.3

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

$- \frac{1}{3} (lim_{t \to \infty} e^{- t^{3}} - lim_{t \to \infty} 1)$

Step 6.3

Since the exponent $- t^{3}$ approaches $- \infty$ , the quantity $e^{- t^{3}}$ approaches $0$ .

$- \frac{1}{3} (0 - lim_{t \to \infty} 1)$

Step 6.4

Evaluate the limit.

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Step 6.4.1

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

$- \frac{1}{3} (0 - 1 \cdot 1)$

Step 6.4.2

Simplify the answer.

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Step 6.4.2.1

Multiply $- 1$ by $1$ .

$- \frac{1}{3} (0 - 1)$

Step 6.4.2.2

Subtract $1$ from $0$ .

$- \frac{1}{3} \cdot - 1$

Step 6.4.2.3

Multiply $- \frac{1}{3} \cdot - 1$ .

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Step 6.4.2.3.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{3})$

Step 6.4.2.3.2

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$