Calculus Examples

$\int_{0}^{3} 21 x^{2} e^{x^{3}} d x$

Step 1

Since $21$ is constant with respect to $x$ , move $21$ out of the integral.

$21 \int_{0}^{3} x^{2} e^{x^{3}} d x$

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 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 2.1.4

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

Anything raised to $0$ is $1$ .

$u_{lower} = 1$

Step 2.4

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

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

Step 2.5

Raise $3$ to the power of $3$ .

$u_{upper} = e^{27}$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = e^{27}$

Step 2.7

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

$21 \int_{1}^{e^{27}} \frac{1}{3} d u_{2}$

Step 3

Apply the constant rule.

$21 (\frac{1}{3} u_{2}]_{1}^{e^{27}})$

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{1}{3}$ and $u_{2}$ .

$21 (\frac{u_{2}}{3}]_{1}^{e^{27}})$

Step 4.2

Evaluate $\frac{u_{2}}{3}$ at $e^{27}$ and at $1$ .

$21 (\frac{e^{27}}{3} - \frac{1}{3})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$21 \frac{e^{27}}{3} + 21 (- \frac{1}{3})$

Step 5.2

Cancel the common factor of $3$ .

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Step 5.2.1

Factor $3$ out of $21$ .

$3 (7) \frac{e^{27}}{3} + 21 (- \frac{1}{3})$

Step 5.2.2

Cancel the common factor.

$3 \cdot 7 \frac{e^{27}}{3} + 21 (- \frac{1}{3})$

Step 5.2.3

Rewrite the expression.

$7 e^{27} + 21 (- \frac{1}{3})$

Step 5.3

Cancel the common factor of $3$ .

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Step 5.3.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$7 e^{27} + 21 (\frac{- 1}{3})$

Step 5.3.2

Factor $3$ out of $21$ .

$7 e^{27} + 3 (7) \frac{- 1}{3}$

Step 5.3.3

Cancel the common factor.

$7 e^{27} + 3 \cdot 7 \frac{- 1}{3}$

Step 5.3.4

Rewrite the expression.

$7 e^{27} + 7 \cdot - 1$

Step 5.4

Multiply $7$ by $- 1$ .

$7 e^{27} - 7$

Step 6

The result can be shown in multiple forms.

Exact Form:

$7 e^{27} - 7$

Decimal Form:

$3724337684205.58 \dots$

Step 7