Calculus Examples

$\int_{0}^{2} 8 x e^{x^{2}} d x$

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int_{0}^{2} x e^{x^{2}} d x$

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 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 2.1.4

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 $u_{2} = e^{x^{2}}$ .

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

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 $u_{2} = e^{x^{2}}$ .

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

Step 2.5

Raise $2$ to the power of $2$ .

$u_{upper} = e^{4}$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = e^{4}$

Step 2.7

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

$8 \int_{1}^{e^{4}} \frac{1}{2} d u_{2}$

Step 3

Apply the constant rule.

$8 (\frac{1}{2} u_{2}]_{1}^{e^{4}})$

Step 4

Simplify the answer.

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Step 4.1

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

$8 (\frac{u_{2}}{2}]_{1}^{e^{4}})$

Step 4.2

Evaluate $\frac{u_{2}}{2}$ at $e^{4}$ and at $1$ .

$8 (\frac{e^{4}}{2} - \frac{1}{2})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$8 \frac{e^{4}}{2} + 8 (- \frac{1}{2})$

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Factor $2$ out of $8$ .

$2 (4) \frac{e^{4}}{2} + 8 (- \frac{1}{2})$

Step 5.2.2

Cancel the common factor.

$2 \cdot 4 \frac{e^{4}}{2} + 8 (- \frac{1}{2})$

Step 5.2.3

Rewrite the expression.

$4 e^{4} + 8 (- \frac{1}{2})$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$4 e^{4} + 8 (\frac{- 1}{2})$

Step 5.3.2

Factor $2$ out of $8$ .

$4 e^{4} + 2 (4) \frac{- 1}{2}$

Step 5.3.3

Cancel the common factor.

$4 e^{4} + 2 \cdot 4 \frac{- 1}{2}$

Step 5.3.4

Rewrite the expression.

$4 e^{4} + 4 \cdot - 1$

Step 5.4

Multiply $4$ by $- 1$ .

$4 e^{4} - 4$

Step 6

The result can be shown in multiple forms.

Exact Form:

$4 e^{4} - 4$

Decimal Form:

$214.39260013 \dots$

Step 7