Calculus Examples

$\int_{1}^{e} 6 x^{2} \ln (x) d x$

Step 1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int_{1}^{e} x^{2} \ln (x) d x$

Step 2

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

$6 (\ln (x) (\frac{1}{3} x^{3})]_{1}^{e} - \int_{1}^{e} \frac{1}{3} x^{3} \frac{1}{x} d x)$

Step 3

Simplify.

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Step 3.1

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

$6 (\ln (x) \frac{x^{3}}{3}]_{1}^{e} - \int_{1}^{e} \frac{1}{3} x^{3} \frac{1}{x} d x)$

Step 3.2

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

$6 (\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \int_{1}^{e} \frac{1}{3} x^{3} \frac{1}{x} d x)$

Step 3.3

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

$6 (\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \int_{1}^{e} \frac{x^{3}}{3} \cdot \frac{1}{x} d x)$

Step 3.4

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

$6 (\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \int_{1}^{e} \frac{x^{3}}{3 x} d x)$

Step 3.5

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

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Step 3.5.1

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

$6 (\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \int_{1}^{e} \frac{x \cdot x^{2}}{3 x} d x)$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $x$ out of $3 x$ .

$6 (\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \int_{1}^{e} \frac{x \cdot x^{2}}{x \cdot 3} d x)$

Step 3.5.2.2

Cancel the common factor.

$6 (\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \int_{1}^{e} \frac{x \cdot x^{2}}{x \cdot 3} d x)$

Step 3.5.2.3

Rewrite the expression.

$6 (\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \int_{1}^{e} \frac{x^{2}}{3} d x)$

Step 4

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

$6 (\frac{\ln (x) x^{3}}{3}]_{1}^{e} - (\frac{1}{3} \int_{1}^{e} x^{2} d x))$

Step 5

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

$6 (\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \frac{1}{3} \frac{1}{3} x^{3}]_{1}^{e})$

Step 6

Simplify the answer.

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Step 6.1

Simplify.

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Step 6.1.1

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

$6 (\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \frac{1}{3} \frac{x^{3}}{3}]_{1}^{e})$

Step 6.1.2

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

$6 (\frac{\ln (x) x^{3}}{3}]_{1}^{e} - \frac{\frac{x^{3}}{3}]_{1}^{e}}{3})$

Step 6.2

Substitute and simplify.

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Step 6.2.1

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

$6 ((\frac{\ln (e) e^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{\frac{x^{3}}{3}]_{1}^{e}}{3})$

Step 6.2.2

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

$6 ((\frac{\ln (e) e^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{(\frac{e^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 6.2.3

Simplify.

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Step 6.2.3.1

One to any power is one.

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1) \cdot 1}{3} - \frac{(\frac{e^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 6.2.3.2

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

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{(\frac{e^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 6.2.3.3

One to any power is one.

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{\frac{e^{3}}{3} - \frac{1}{3}}{3})$

Step 6.2.3.4

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

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - (\frac{3}{3} \cdot \frac{\frac{e^{3}}{3} - \frac{1}{3}}{3}))$

Step 6.2.3.5

Combine.

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{3 (\frac{e^{3}}{3} - \frac{1}{3})}{3 \cdot 3})$

Step 6.2.3.6

Apply the distributive property.

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{3 \frac{e^{3}}{3} + 3 (- \frac{1}{3})}{3 \cdot 3})$

Step 6.2.3.7

Cancel the common factor of $3$ .

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Step 6.2.3.7.1

Cancel the common factor.

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{3 \frac{e^{3}}{3} + 3 (- \frac{1}{3})}{3 \cdot 3})$

Step 6.2.3.7.2

Rewrite the expression.

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{e^{3} + 3 (- \frac{1}{3})}{3 \cdot 3})$

Step 6.2.3.8

Multiply $- 1$ by $3$ .

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{e^{3} - 3 (\frac{1}{3})}{3 \cdot 3})$

Step 6.2.3.9

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

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{e^{3} + \frac{- 3}{3}}{3 \cdot 3})$

Step 6.2.3.10

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

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Step 6.2.3.10.1

Factor $3$ out of $- 3$ .

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{e^{3} + \frac{3 \cdot - 1}{3}}{3 \cdot 3})$

Step 6.2.3.10.2

Cancel the common factors.

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Step 6.2.3.10.2.1

Factor $3$ out of $3$ .

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{e^{3} + \frac{3 \cdot - 1}{3 (1)}}{3 \cdot 3})$

Step 6.2.3.10.2.2

Cancel the common factor.

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{e^{3} + \frac{3 \cdot - 1}{3 \cdot 1}}{3 \cdot 3})$

Step 6.2.3.10.2.3

Rewrite the expression.

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{e^{3} + \frac{- 1}{1}}{3 \cdot 3})$

Step 6.2.3.10.2.4

Divide $- 1$ by $1$ .

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{e^{3} - 1}{3 \cdot 3})$

Step 6.2.3.11

Multiply $3$ by $3$ .

$6 (\frac{\ln (e) e^{3}}{3} - \frac{\ln (1)}{3} - \frac{e^{3} - 1}{9})$

Step 6.2.3.12

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

$6 (\frac{\ln (e) e^{3}}{3} \cdot \frac{3}{3} - \frac{e^{3} - 1}{9} - \frac{\ln (1)}{3})$

Step 6.2.3.13

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.3.13.1

Multiply $\frac{\ln (e) e^{3}}{3}$ by $\frac{3}{3}$ .

$6 (\frac{\ln (e) e^{3} \cdot 3}{3 \cdot 3} - \frac{e^{3} - 1}{9} - \frac{\ln (1)}{3})$

Step 6.2.3.13.2

Multiply $3$ by $3$ .

$6 (\frac{\ln (e) e^{3} \cdot 3}{9} - \frac{e^{3} - 1}{9} - \frac{\ln (1)}{3})$

Step 6.2.3.14

Combine the numerators over the common denominator.

$6 (\frac{\ln (e) e^{3} \cdot 3 - (e^{3} - 1)}{9} - \frac{\ln (1)}{3})$

Step 6.2.3.15

Move $3$ to the left of $\ln (e) e^{3}$ .

$6 (\frac{3 \ln (e) e^{3} - (e^{3} - 1)}{9} - \frac{\ln (1)}{3})$

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

Simplify the numerator.

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Step 7.1.1.1

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

$6 (\frac{3 \cdot 1 e^{3} - (e^{3} - 1)}{9} - \frac{\ln (1)}{3})$

Step 7.1.1.2

Multiply $3$ by $1$ .

$6 (\frac{3 e^{3} - (e^{3} - 1)}{9} - \frac{\ln (1)}{3})$

Step 7.1.1.3

Apply the distributive property.

$6 (\frac{3 e^{3} - e^{3} - - 1}{9} - \frac{\ln (1)}{3})$

Step 7.1.1.4

Multiply $- 1$ by $- 1$ .

$6 (\frac{3 e^{3} - e^{3} + 1}{9} - \frac{\ln (1)}{3})$

Step 7.1.1.5

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

$6 (\frac{2 e^{3} + 1}{9} - \frac{\ln (1)}{3})$

Step 7.1.2

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

$6 (\frac{2 e^{3} + 1}{9} - \frac{0}{3})$

Step 7.1.3

Divide $0$ by $3$ .

$6 (\frac{2 e^{3} + 1}{9} - 0)$

Step 7.1.4

Multiply $- 1$ by $0$ .

$6 (\frac{2 e^{3} + 1}{9} + 0)$

Step 7.2

Add $\frac{2 e^{3} + 1}{9}$ and $0$ .

$6 \frac{2 e^{3} + 1}{9}$

Step 7.3

Cancel the common factor of $3$ .

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Step 7.3.1

Factor $3$ out of $6$ .

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

Step 7.3.2

Factor $3$ out of $9$ .

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

Step 7.3.3

Cancel the common factor.

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

Step 7.3.4

Rewrite the expression.

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

Step 7.4

Combine $2$ and $\frac{2 e^{3} + 1}{3}$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$27.44738256 \dots$