Calculus Examples

$\int_{2}^{4} 6 x \ln (x) d x$

Step 1

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

$6 \int_{2}^{4} x \ln (x) d x$

Step 2

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

$6 (\ln (x) (\frac{1}{2} x^{2})]_{2}^{4} - \int_{2}^{4} \frac{1}{2} x^{2} \frac{1}{x} d x)$

Step 3

Simplify.

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Step 3.1

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

$6 (\ln (x) \frac{x^{2}}{2}]_{2}^{4} - \int_{2}^{4} \frac{1}{2} x^{2} \frac{1}{x} d x)$

Step 3.2

Combine $\ln (x)$ and $\frac{x^{2}}{2}$ .

$6 (\frac{\ln (x) x^{2}}{2}]_{2}^{4} - \int_{2}^{4} \frac{1}{2} x^{2} \frac{1}{x} d x)$

Step 3.3

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

$6 (\frac{\ln (x) x^{2}}{2}]_{2}^{4} - \int_{2}^{4} \frac{x^{2}}{2} \cdot \frac{1}{x} d x)$

Step 3.4

Multiply $\frac{x^{2}}{2}$ by $\frac{1}{x}$ .

$6 (\frac{\ln (x) x^{2}}{2}]_{2}^{4} - \int_{2}^{4} \frac{x^{2}}{2 x} d x)$

Step 3.5

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

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Step 3.5.1

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

$6 (\frac{\ln (x) x^{2}}{2}]_{2}^{4} - \int_{2}^{4} \frac{x \cdot x}{2 x} d x)$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $x$ out of $2 x$ .

$6 (\frac{\ln (x) x^{2}}{2}]_{2}^{4} - \int_{2}^{4} \frac{x \cdot x}{x \cdot 2} d x)$

Step 3.5.2.2

Cancel the common factor.

$6 (\frac{\ln (x) x^{2}}{2}]_{2}^{4} - \int_{2}^{4} \frac{x \cdot x}{x \cdot 2} d x)$

Step 3.5.2.3

Rewrite the expression.

$6 (\frac{\ln (x) x^{2}}{2}]_{2}^{4} - \int_{2}^{4} \frac{x}{2} d x)$

Step 4

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

$6 (\frac{\ln (x) x^{2}}{2}]_{2}^{4} - (\frac{1}{2} \int_{2}^{4} x d x))$

Step 5

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

$6 (\frac{\ln (x) x^{2}}{2}]_{2}^{4} - \frac{1}{2} \frac{1}{2} x^{2}]_{2}^{4})$

Step 6

Simplify the answer.

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Step 6.1

Simplify.

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Step 6.1.1

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

$6 (\frac{\ln (x) x^{2}}{2}]_{2}^{4} - \frac{1}{2} \frac{x^{2}}{2}]_{2}^{4})$

Step 6.1.2

Combine $\frac{x^{2}}{2}]_{2}^{4}$ and $\frac{1}{2}$ .

$6 (\frac{\ln (x) x^{2}}{2}]_{2}^{4} - \frac{\frac{x^{2}}{2}]_{2}^{4}}{2})$

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $\frac{\ln (x) x^{2}}{2}$ at $4$ and at $2$ .

$6 ((\frac{\ln (4) \cdot 4^{2}}{2}) - \frac{\ln (2) \cdot 2^{2}}{2} - \frac{\frac{x^{2}}{2}]_{2}^{4}}{2})$

Step 6.2.2

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $2$ .

$6 ((\frac{\ln (4) \cdot 4^{2}}{2}) - \frac{\ln (2) \cdot 2^{2}}{2} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Raise $4$ to the power of $2$ .

$6 (\frac{\ln (4) \cdot 16}{2} - \frac{\ln (2) \cdot 2^{2}}{2} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.2

Move $16$ to the left of $\ln (4)$ .

$6 (\frac{16 \cdot \ln (4)}{2} - \frac{\ln (2) \cdot 2^{2}}{2} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.3

Cancel the common factor of $16$ and $2$ .

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Step 6.2.3.3.1

Factor $2$ out of $16 \ln (4)$ .

$6 (\frac{2 (8 \ln (4))}{2} - \frac{\ln (2) \cdot 2^{2}}{2} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.3.2

Cancel the common factors.

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Step 6.2.3.3.2.1

Factor $2$ out of $2$ .

$6 (\frac{2 (8 \ln (4))}{2 (1)} - \frac{\ln (2) \cdot 2^{2}}{2} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.3.2.2

Cancel the common factor.

$6 (\frac{2 (8 \ln (4))}{2 \cdot 1} - \frac{\ln (2) \cdot 2^{2}}{2} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.3.2.3

Rewrite the expression.

$6 (\frac{8 \ln (4)}{1} - \frac{\ln (2) \cdot 2^{2}}{2} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.3.2.4

Divide $8 \ln (4)$ by $1$ .

$6 (8 \ln (4) - \frac{\ln (2) \cdot 2^{2}}{2} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.4

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

$6 (8 \ln (4) - \frac{\ln (2) \cdot 4}{2} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.5

Move $4$ to the left of $\ln (2)$ .

$6 (8 \ln (4) - \frac{4 \cdot \ln (2)}{2} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.6

Cancel the common factor of $4$ and $2$ .

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Step 6.2.3.6.1

Factor $2$ out of $4 \ln (2)$ .

$6 (8 \ln (4) - \frac{2 (2 \ln (2))}{2} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.6.2

Cancel the common factors.

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Step 6.2.3.6.2.1

Factor $2$ out of $2$ .

$6 (8 \ln (4) - \frac{2 (2 \ln (2))}{2 (1)} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.6.2.2

Cancel the common factor.

$6 (8 \ln (4) - \frac{2 (2 \ln (2))}{2 \cdot 1} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.6.2.3

Rewrite the expression.

$6 (8 \ln (4) - \frac{2 \ln (2)}{1} - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.6.2.4

Divide $2 \ln (2)$ by $1$ .

$6 (8 \ln (4) - (2 \ln (2)) - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.7

Multiply $2$ by $- 1$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{(\frac{4^{2}}{2}) - \frac{2^{2}}{2}}{2})$

Step 6.2.3.8

Raise $4$ to the power of $2$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{\frac{16}{2} - \frac{2^{2}}{2}}{2})$

Step 6.2.3.9

Cancel the common factor of $16$ and $2$ .

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Step 6.2.3.9.1

Factor $2$ out of $16$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{\frac{2 \cdot 8}{2} - \frac{2^{2}}{2}}{2})$

Step 6.2.3.9.2

Cancel the common factors.

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Step 6.2.3.9.2.1

Factor $2$ out of $2$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{\frac{2 \cdot 8}{2 (1)} - \frac{2^{2}}{2}}{2})$

Step 6.2.3.9.2.2

Cancel the common factor.

$6 (8 \ln (4) - 2 \ln (2) - \frac{\frac{2 \cdot 8}{2 \cdot 1} - \frac{2^{2}}{2}}{2})$

Step 6.2.3.9.2.3

Rewrite the expression.

$6 (8 \ln (4) - 2 \ln (2) - \frac{\frac{8}{1} - \frac{2^{2}}{2}}{2})$

Step 6.2.3.9.2.4

Divide $8$ by $1$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{8 - \frac{2^{2}}{2}}{2})$

Step 6.2.3.10

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

$6 (8 \ln (4) - 2 \ln (2) - \frac{8 - \frac{4}{2}}{2})$

Step 6.2.3.11

Cancel the common factor of $4$ and $2$ .

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Step 6.2.3.11.1

Factor $2$ out of $4$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{8 - \frac{2 \cdot 2}{2}}{2})$

Step 6.2.3.11.2

Cancel the common factors.

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Step 6.2.3.11.2.1

Factor $2$ out of $2$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{8 - \frac{2 \cdot 2}{2 (1)}}{2})$

Step 6.2.3.11.2.2

Cancel the common factor.

$6 (8 \ln (4) - 2 \ln (2) - \frac{8 - \frac{2 \cdot 2}{2 \cdot 1}}{2})$

Step 6.2.3.11.2.3

Rewrite the expression.

$6 (8 \ln (4) - 2 \ln (2) - \frac{8 - \frac{2}{1}}{2})$

Step 6.2.3.11.2.4

Divide $2$ by $1$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{8 - 1 \cdot 2}{2})$

Step 6.2.3.12

Multiply $- 1$ by $2$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{8 - 2}{2})$

Step 6.2.3.13

Subtract $2$ from $8$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{6}{2})$

Step 6.2.3.14

Cancel the common factor of $6$ and $2$ .

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Step 6.2.3.14.1

Factor $2$ out of $6$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{2 \cdot 3}{2})$

Step 6.2.3.14.2

Cancel the common factors.

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Step 6.2.3.14.2.1

Factor $2$ out of $2$ .

$6 (8 \ln (4) - 2 \ln (2) - \frac{2 \cdot 3}{2 (1)})$

Step 6.2.3.14.2.2

Cancel the common factor.

$6 (8 \ln (4) - 2 \ln (2) - \frac{2 \cdot 3}{2 \cdot 1})$

Step 6.2.3.14.2.3

Rewrite the expression.

$6 (8 \ln (4) - 2 \ln (2) - \frac{3}{1})$

Step 6.2.3.14.2.4

Divide $3$ by $1$ .

$6 (8 \ln (4) - 2 \ln (2) - 1 \cdot 3)$

Step 6.2.3.15

Multiply $- 1$ by $3$ .

$6 (8 \ln (4) - 2 \ln (2) - 3)$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$6 (8 \ln (4)) + 6 (- 2 \ln (2)) + 6 \cdot - 3$

Step 7.2

Simplify.

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Step 7.2.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$6 (8 \ln (2^{2})) + 6 (- 2 \ln (2)) + 6 \cdot - 3$

Step 7.2.2

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

$6 (8 (2 \ln (2))) + 6 (- 2 \ln (2)) + 6 \cdot - 3$

Step 7.2.3

Multiply $- 2$ by $6$ .

$6 (8 (2 \ln (2))) - 12 \ln (2) + 6 \cdot - 3$

Step 7.2.4

Multiply $6$ by $- 3$ .

$6 (8 (2 \ln (2))) - 12 \ln (2) - 18$

Step 7.3

Simplify each term.

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Step 7.3.1

Multiply $2$ by $8$ .

$6 (16 \ln (2)) - 12 \ln (2) - 18$

Step 7.3.2

Multiply $16$ by $6$ .

$96 \ln (2) - 12 \ln (2) - 18$

Step 7.4

Subtract $12 \ln (2)$ from $96 \ln (2)$ .

$84 \ln (2) - 18$

Step 8

The result can be shown in multiple forms.

Exact Form:

$84 \ln (2) - 18$

Decimal Form:

$40.22436316 \dots$