Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{2}$ .

$\ln (x) (\frac{1}{3} x^{3})]_{1}^{e} - \int_{1}^{e} \frac{1}{3} x^{3} \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$\ln (x) \frac{x^{3}}{3}]_{1}^{e} - \int_{1}^{e} \frac{1}{3} x^{3} \frac{1}{x} d x$

Step 2.2

Combine $\ln (x)$ and $\frac{x^{3}}{3}$ .

$\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \int_{1}^{e} \frac{1}{3} x^{3} \frac{1}{x} d x$

Step 3

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{\ln (x) x^{3}}{3}]_{1}^{e} - (\frac{1}{3} \int_{1}^{e} x^{3} \frac{1}{x} d x)$

Step 4

Simplify.

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Step 4.1

Combine $x^{3}$ and $\frac{1}{x}$ .

$\frac{\ln (x) x^{3}}{3}]_{1}^{e} - (\frac{1}{3} \int_{1}^{e} \frac{x^{3}}{x} d x)$

Step 4.2

Cancel the common factor of $x^{3}$ and $x$ .

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Step 4.2.1

Factor $x$ out of $x^{3}$ .

$\frac{\ln (x) x^{3}}{3}]_{1}^{e} - (\frac{1}{3} \int_{1}^{e} \frac{x \cdot x^{2}}{x} d x)$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Raise $x$ to the power of $1$ .

$\frac{\ln (x) x^{3}}{3}]_{1}^{e} - (\frac{1}{3} \int_{1}^{e} \frac{x \cdot x^{2}}{x^{1}} d x)$

Step 4.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{\ln (x) x^{3}}{3}]_{1}^{e} - (\frac{1}{3} \int_{1}^{e} \frac{x \cdot x^{2}}{x \cdot 1} d x)$

Step 4.2.2.3

Cancel the common factor.

$\frac{\ln (x) x^{3}}{3}]_{1}^{e} - (\frac{1}{3} \int_{1}^{e} \frac{x \cdot x^{2}}{x \cdot 1} d x)$

Step 4.2.2.4

Rewrite the expression.

$\frac{\ln (x) x^{3}}{3}]_{1}^{e} - (\frac{1}{3} \int_{1}^{e} \frac{x^{2}}{1} d x)$

Step 4.2.2.5

Divide $x^{2}$ by $1$ .

$\frac{\ln (x) x^{3}}{3}]_{1}^{e} - (\frac{1}{3} \int_{1}^{e} x^{2} d x)$

$\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \frac{1}{3} \int_{1}^{e} x^{2} d x$

Step 5

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

$\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \frac{1}{3} \frac{1}{3} x^{3}]_{1}^{e}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{\ln (x) x^{3}}{3}$ at $e$ and at $1$ .

$(\frac{\ln (e) e^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{1}{3} \frac{1}{3} x^{3}]_{1}^{e}$

Step 6.2

Evaluate $\frac{1}{3} x^{3}$ at $e$ and at $1$ .

$(\frac{\ln (e) e^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{1}{3} ((\frac{1}{3} e^{3}) - \frac{1}{3} \cdot 1^{3})$

Step 6.3

Simplify.

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Step 6.3.1

One to any power is one.

$\frac{\ln (e) e^{3}}{3} - \frac{\ln (1) \cdot 1}{3} - \frac{1}{3} ((\frac{1}{3} e^{3}) - \frac{1}{3} \cdot 1^{3})$

Step 6.3.2

Multiply $\ln (1)$ by $1$ .

$\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{1}{3} ((\frac{1}{3} e^{3}) - \frac{1}{3} \cdot 1^{3})$

Step 6.3.3

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

$\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{1}{3} (\frac{e^{3}}{3} - \frac{1}{3} \cdot 1^{3})$

Step 6.3.4

One to any power is one.

$\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{1}{3} (\frac{e^{3}}{3} - \frac{1}{3} \cdot 1)$

Step 6.3.5

Multiply $- 1$ by $1$ .

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

Step 6.3.6

To write $- \frac{1}{3} (\frac{e^{3}}{3} - \frac{1}{3})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\ln (e) e^{3}}{3} - \frac{1}{3} (\frac{e^{3}}{3} - \frac{1}{3}) \cdot \frac{3}{3} - \frac{\ln (1)}{3}$

Step 6.3.7

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

$\frac{\ln (e) e^{3}}{3} + \frac{- \frac{1}{3} (\frac{e^{3}}{3} - \frac{1}{3}) \cdot 3}{3} - \frac{\ln (1)}{3}$

Step 6.3.8

Combine the numerators over the common denominator.

$\frac{\ln (e) e^{3} - \frac{1}{3} (\frac{e^{3}}{3} - \frac{1}{3}) \cdot 3}{3} - \frac{\ln (1)}{3}$

Step 6.3.9

Multiply $3$ by $- 1$ .

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

Step 6.3.10

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

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

Step 6.3.11

Cancel the common factor of $- 3$ and $3$ .

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Step 6.3.11.1

Factor $3$ out of $- 3$ .

$\frac{\ln (e) e^{3} + \frac{3 \cdot - 1}{3} (\frac{e^{3}}{3} - \frac{1}{3})}{3} - \frac{\ln (1)}{3}$

Step 6.3.11.2

Cancel the common factors.

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Step 6.3.11.2.1

Factor $3$ out of $3$ .

$\frac{\ln (e) e^{3} + \frac{3 \cdot - 1}{3 (1)} (\frac{e^{3}}{3} - \frac{1}{3})}{3} - \frac{\ln (1)}{3}$

Step 6.3.11.2.2

Cancel the common factor.

$\frac{\ln (e) e^{3} + \frac{3 \cdot - 1}{3 \cdot 1} (\frac{e^{3}}{3} - \frac{1}{3})}{3} - \frac{\ln (1)}{3}$

Step 6.3.11.2.3

Rewrite the expression.

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

Step 6.3.11.2.4

Divide $- 1$ by $1$ .

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

Step 7

Simplify.

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Step 7.1

Combine the numerators over the common denominator.

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

Step 7.2

Simplify each term.

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Step 7.2.1

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

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

Step 7.2.2

Multiply $e^{3}$ by $1$ .

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

Step 7.2.3

Apply the distributive property.

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

Step 7.2.4

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

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Step 7.2.4.1

Multiply $- 1$ by $- 1$ .

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

Step 7.2.4.2

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

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

Step 7.2.5

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

$\frac{e^{3} - \frac{e^{3}}{3} + \frac{1}{3} - 0}{3}$

Step 7.2.6

Multiply $- 1$ by $0$ .

$\frac{e^{3} - \frac{e^{3}}{3} + \frac{1}{3} + 0}{3}$

Step 7.3

Combine the numerators over the common denominator.

$\frac{e^{3} + \frac{- e^{3}}{3} + \frac{1}{3}}{3}$

Step 7.4

Move the negative in front of the fraction.

$\frac{e^{3} - \frac{e^{3}}{3} + \frac{1}{3}}{3}$

Step 7.5

To write $e^{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 7.6

Combine $e^{3}$ and $\frac{3}{3}$ .

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

Step 7.7

Combine the numerators over the common denominator.

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

Step 7.8

Combine the numerators over the common denominator.

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

Step 7.9

Move $3$ to the left of $e^{3}$ .

$\frac{\frac{3 e^{3} - e^{3} + 1}{3}}{3}$

Step 7.10

Subtract $e^{3}$ from $3 e^{3}$ .

$\frac{\frac{2 e^{3} + 1}{3}}{3}$

Step 7.11

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 e^{3} + 1}{3} \cdot \frac{1}{3}$

Step 7.12

Multiply $\frac{2 e^{3} + 1}{3} \cdot \frac{1}{3}$ .

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Step 7.12.1

Multiply $\frac{2 e^{3} + 1}{3}$ by $\frac{1}{3}$ .

$\frac{2 e^{3} + 1}{3 \cdot 3}$

Step 7.12.2

Multiply $3$ by $3$ .

$\frac{2 e^{3} + 1}{9}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2 e^{3} + 1}{9}$

Decimal Form:

$4.57456376 \dots$