Calculus Examples

$\int_{1}^{3} 9 r^{3} \ln (r) d r$

Step 1

Since $9$ is constant with respect to $r$ , move $9$ out of the integral.

$9 \int_{1}^{3} r^{3} \ln (r) d r$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (r)$ and $d v = r^{3}$ .

$9 (\ln (r) (\frac{1}{4} r^{4})]_{1}^{3} - \int_{1}^{3} \frac{1}{4} r^{4} \frac{1}{r} d r)$

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{4}$ and $r^{4}$ .

$9 (\ln (r) \frac{r^{4}}{4}]_{1}^{3} - \int_{1}^{3} \frac{1}{4} r^{4} \frac{1}{r} d r)$

Step 3.2

Combine $\ln (r)$ and $\frac{r^{4}}{4}$ .

$9 (\frac{\ln (r) r^{4}}{4}]_{1}^{3} - \int_{1}^{3} \frac{1}{4} r^{4} \frac{1}{r} d r)$

Step 3.3

Combine $\frac{1}{4}$ and $r^{4}$ .

$9 (\frac{\ln (r) r^{4}}{4}]_{1}^{3} - \int_{1}^{3} \frac{r^{4}}{4} \cdot \frac{1}{r} d r)$

Step 3.4

Multiply $\frac{r^{4}}{4}$ by $\frac{1}{r}$ .

$9 (\frac{\ln (r) r^{4}}{4}]_{1}^{3} - \int_{1}^{3} \frac{r^{4}}{4 r} d r)$

Step 3.5

Cancel the common factor of $r^{4}$ and $r$ .

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Step 3.5.1

Factor $r$ out of $r^{4}$ .

$9 (\frac{\ln (r) r^{4}}{4}]_{1}^{3} - \int_{1}^{3} \frac{r \cdot r^{3}}{4 r} d r)$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $r$ out of $4 r$ .

$9 (\frac{\ln (r) r^{4}}{4}]_{1}^{3} - \int_{1}^{3} \frac{r \cdot r^{3}}{r \cdot 4} d r)$

Step 3.5.2.2

Cancel the common factor.

$9 (\frac{\ln (r) r^{4}}{4}]_{1}^{3} - \int_{1}^{3} \frac{r \cdot r^{3}}{r \cdot 4} d r)$

Step 3.5.2.3

Rewrite the expression.

$9 (\frac{\ln (r) r^{4}}{4}]_{1}^{3} - \int_{1}^{3} \frac{r^{3}}{4} d r)$

Step 4

Since $\frac{1}{4}$ is constant with respect to $r$ , move $\frac{1}{4}$ out of the integral.

$9 (\frac{\ln (r) r^{4}}{4}]_{1}^{3} - (\frac{1}{4} \int_{1}^{3} r^{3} d r))$

Step 5

By the Power Rule, the integral of $r^{3}$ with respect to $r$ is $\frac{1}{4} r^{4}$ .

$9 (\frac{\ln (r) r^{4}}{4}]_{1}^{3} - \frac{1}{4} \frac{1}{4} r^{4}]_{1}^{3})$

Step 6

Simplify the answer.

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Step 6.1

Simplify.

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Step 6.1.1

Combine $\frac{1}{4}$ and $r^{4}$ .

$9 (\frac{\ln (r) r^{4}}{4}]_{1}^{3} - \frac{1}{4} \frac{r^{4}}{4}]_{1}^{3})$

Step 6.1.2

Combine $\frac{r^{4}}{4}]_{1}^{3}$ and $\frac{1}{4}$ .

$9 (\frac{\ln (r) r^{4}}{4}]_{1}^{3} - \frac{\frac{r^{4}}{4}]_{1}^{3}}{4})$

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $\frac{\ln (r) r^{4}}{4}$ at $3$ and at $1$ .

$9 ((\frac{\ln (3) \cdot 3^{4}}{4}) - \frac{\ln (1) \cdot 1^{4}}{4} - \frac{\frac{r^{4}}{4}]_{1}^{3}}{4})$

Step 6.2.2

Evaluate $\frac{r^{4}}{4}$ at $3$ and at $1$ .

$9 ((\frac{\ln (3) \cdot 3^{4}}{4}) - \frac{\ln (1) \cdot 1^{4}}{4} - \frac{(\frac{3^{4}}{4}) - \frac{1^{4}}{4}}{4})$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Raise $3$ to the power of $4$ .

$9 (\frac{\ln (3) \cdot 81}{4} - \frac{\ln (1) \cdot 1^{4}}{4} - \frac{(\frac{3^{4}}{4}) - \frac{1^{4}}{4}}{4})$

Step 6.2.3.2

Move $81$ to the left of $\ln (3)$ .

$9 (\frac{81 \cdot \ln (3)}{4} - \frac{\ln (1) \cdot 1^{4}}{4} - \frac{(\frac{3^{4}}{4}) - \frac{1^{4}}{4}}{4})$

Step 6.2.3.3

One to any power is one.

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1) \cdot 1}{4} - \frac{(\frac{3^{4}}{4}) - \frac{1^{4}}{4}}{4})$

Step 6.2.3.4

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

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{(\frac{3^{4}}{4}) - \frac{1^{4}}{4}}{4})$

Step 6.2.3.5

Raise $3$ to the power of $4$ .

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{\frac{81}{4} - \frac{1^{4}}{4}}{4})$

Step 6.2.3.6

One to any power is one.

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{\frac{81}{4} - \frac{1}{4}}{4})$

Step 6.2.3.7

Combine the numerators over the common denominator.

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{\frac{81 - 1}{4}}{4})$

Step 6.2.3.8

Subtract $1$ from $81$ .

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{\frac{80}{4}}{4})$

Step 6.2.3.9

Cancel the common factor of $80$ and $4$ .

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Step 6.2.3.9.1

Factor $4$ out of $80$ .

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{\frac{4 \cdot 20}{4}}{4})$

Step 6.2.3.9.2

Cancel the common factors.

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Step 6.2.3.9.2.1

Factor $4$ out of $4$ .

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{\frac{4 \cdot 20}{4 (1)}}{4})$

Step 6.2.3.9.2.2

Cancel the common factor.

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{\frac{4 \cdot 20}{4 \cdot 1}}{4})$

Step 6.2.3.9.2.3

Rewrite the expression.

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{\frac{20}{1}}{4})$

Step 6.2.3.9.2.4

Divide $20$ by $1$ .

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{20}{4})$

Step 6.2.3.10

Cancel the common factor of $20$ and $4$ .

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Step 6.2.3.10.1

Factor $4$ out of $20$ .

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{4 \cdot 5}{4})$

Step 6.2.3.10.2

Cancel the common factors.

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Step 6.2.3.10.2.1

Factor $4$ out of $4$ .

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{4 \cdot 5}{4 (1)})$

Step 6.2.3.10.2.2

Cancel the common factor.

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{4 \cdot 5}{4 \cdot 1})$

Step 6.2.3.10.2.3

Rewrite the expression.

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - \frac{5}{1})$

Step 6.2.3.10.2.4

Divide $5$ by $1$ .

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - 1 \cdot 5)$

Step 6.2.3.11

Multiply $- 1$ by $5$ .

$9 (\frac{81 \ln (3)}{4} - \frac{\ln (1)}{4} - 5)$

Step 7

Simplify.

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Step 7.1

Combine the numerators over the common denominator.

$9 (- 5 + \frac{81 \ln (3) - \ln (1)}{4})$

Step 7.2

Simplify each term.

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Step 7.2.1

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

$9 (- 5 + \frac{81 \ln (3) - 0}{4})$

Step 7.2.2

Multiply $- 1$ by $0$ .

$9 (- 5 + \frac{81 \ln (3) + 0}{4})$

Step 7.3

Add $81 \ln (3)$ and $0$ .

$9 (- 5 + \frac{81 \ln (3)}{4})$

Step 7.4

Apply the distributive property.

$9 \cdot - 5 + 9 \frac{81 \ln (3)}{4}$

Step 7.5

Multiply $9$ by $- 5$ .

$- 45 + 9 \frac{81 \ln (3)}{4}$

Step 7.6

Multiply $9 \frac{81 \ln (3)}{4}$ .

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Step 7.6.1

Combine $9$ and $\frac{81 \ln (3)}{4}$ .

$- 45 + \frac{9 (81 \ln (3))}{4}$

Step 7.6.2

Multiply $81$ by $9$ .

$- 45 + \frac{729 \ln (3)}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- 45 + \frac{729 \ln (3)}{4}$

Decimal Form:

$155.22208960 \dots$