Calculus Examples

$\int_{1}^{e^{8}} \frac{\ln^{2} (x^{2})}{x} d x$

Step 1

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

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Step 1.1

Let $u_{2} = \ln (x^{2})$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\ln (x^{2})$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [x^{2}]$

Step 1.1.2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 1.1.2.3

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

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

Step 1.1.3

Differentiate using the Power Rule.

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Step 1.1.3.1

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

$\frac{1}{x^{2}} (2 x)$

Step 1.1.3.2

Simplify terms.

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Step 1.1.3.2.1

Combine $2$ and $\frac{1}{x^{2}}$ .

$\frac{2}{x^{2}} x$

Step 1.1.3.2.2

Combine $\frac{2}{x^{2}}$ and $x$ .

$\frac{2 x}{x^{2}}$

Step 1.1.3.2.3

Cancel the common factor of $x$ and $x^{2}$ .

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Step 1.1.3.2.3.1

Factor $x$ out of $2 x$ .

$\frac{x \cdot 2}{x^{2}}$

Step 1.1.3.2.3.2

Cancel the common factors.

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Step 1.1.3.2.3.2.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot 2}{x \cdot x}$

Step 1.1.3.2.3.2.2

Cancel the common factor.

$\frac{x \cdot 2}{x \cdot x}$

Step 1.1.3.2.3.2.3

Rewrite the expression.

$\frac{2}{x}$

Step 1.2

Substitute the lower limit in for $x$ in $u_{2} = \ln (x^{2})$ .

$u_{lower} = \ln (1^{2})$

Step 1.3

Simplify.

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Step 1.3.1

One to any power is one.

$u_{lower} = \ln (1)$

Step 1.3.2

The natural logarithm of $1$ is $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u_{2} = \ln (x^{2})$ .

$u_{upper} = \ln ({(e^{8})}^{2})$

Step 1.5

Simplify.

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Step 1.5.1

Multiply the exponents in ${(e^{8})}^{2}$ .

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Step 1.5.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$u_{upper} = \ln (e^{8 \cdot 2})$

Step 1.5.1.2

Multiply $8$ by $2$ .

$u_{upper} = \ln (e^{16})$

Step 1.5.2

Use logarithm rules to move $16$ out of the exponent.

$u_{upper} = 16 \ln (e)$

Step 1.5.3

The natural logarithm of $e$ is $1$ .

$u_{upper} = 16 \cdot 1$

Step 1.5.4

Multiply $16$ by $1$ .

$u_{upper} = 16$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 16$

Step 1.7

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

$\int_{0}^{16} \frac{1}{2} {u_{2}}^{2} d u_{2}$

Step 2

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{0}^{16} {u_{2}}^{2} d u_{2}$

Step 3

By the Power Rule, the integral of ${u_{2}}^{2}$ with respect to $u_{2}$ is $\frac{1}{3} {u_{2}}^{3}$ .

$\frac{1}{2} \frac{1}{3} {u_{2}}^{3}]_{0}^{16}$

Step 4

Combine fractions.

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Step 4.1

Evaluate $\frac{1}{3} {u_{2}}^{3}$ at $16$ and at $0$ .

$\frac{1}{2} ((\frac{1}{3} \cdot 16^{3}) - \frac{1}{3} \cdot 0^{3})$

Step 4.2

Raise $16$ to the power of $3$ .

$\frac{1}{2} (\frac{1}{3} \cdot 4096 - \frac{1}{3} \cdot 0^{3})$

Step 4.3

Combine $\frac{1}{3}$ and $4096$ .

$\frac{1}{2} (\frac{4096}{3} - \frac{1}{3} \cdot 0^{3})$

Step 4.4

Simplify the expression.

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Step 4.4.1

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

$\frac{1}{2} (\frac{4096}{3} - \frac{1}{3} \cdot 0)$

Step 4.4.2

Simplify.

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Step 4.4.2.1

Multiply $0$ by $- 1$ .

$\frac{1}{2} (\frac{4096}{3} + 0 (\frac{1}{3}))$

Step 4.4.2.2

Multiply $0$ by $\frac{1}{3}$ .

$\frac{1}{2} (\frac{4096}{3} + 0)$

Step 4.4.3

Add $\frac{4096}{3}$ and $0$ .

$\frac{1}{2} \cdot \frac{4096}{3}$

Step 4.4.4

Simplify.

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Step 4.4.4.1

Multiply $\frac{1}{2}$ by $\frac{4096}{3}$ .

$\frac{4096}{2 \cdot 3}$

Step 4.4.4.2

Multiply $2$ by $3$ .

$\frac{4096}{6}$

Step 4.4.4.3

Cancel the common factor of $4096$ and $6$ .

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Step 4.4.4.3.1

Factor $2$ out of $4096$ .

$\frac{2 (2048)}{6}$

Step 4.4.4.3.2

Cancel the common factors.

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Step 4.4.4.3.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 2048}{2 \cdot 3}$

Step 4.4.4.3.2.2

Cancel the common factor.

$\frac{2 \cdot 2048}{2 \cdot 3}$

Step 4.4.4.3.2.3

Rewrite the expression.

$\frac{2048}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2048}{3}$

Decimal Form:

$682.\overline{6}$

Mixed Number Form:

$682 \frac{2}{3}$