Calculus Examples

$\int_{1}^{5} w^{2} \ln (w) d w$

Step 1

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

$\ln (w) (\frac{1}{3} w^{3})]_{1}^{5} - \int_{1}^{5} \frac{1}{3} w^{3} \frac{1}{w} d w$

Step 2

Simplify.

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Step 2.1

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

$\ln (w) \frac{w^{3}}{3}]_{1}^{5} - \int_{1}^{5} \frac{1}{3} w^{3} \frac{1}{w} d w$

Step 2.2

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

$\frac{\ln (w) w^{3}}{3}]_{1}^{5} - \int_{1}^{5} \frac{1}{3} w^{3} \frac{1}{w} d w$

Step 3

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

$\frac{\ln (w) w^{3}}{3}]_{1}^{5} - (\frac{1}{3} \int_{1}^{5} w^{3} \frac{1}{w} d w)$

Step 4

Simplify.

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Step 4.1

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

$\frac{\ln (w) w^{3}}{3}]_{1}^{5} - (\frac{1}{3} \int_{1}^{5} \frac{w^{3}}{w} d w)$

Step 4.2

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

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Step 4.2.1

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

$\frac{\ln (w) w^{3}}{3}]_{1}^{5} - (\frac{1}{3} \int_{1}^{5} \frac{w \cdot w^{2}}{w} d w)$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

$\frac{\ln (w) w^{3}}{3}]_{1}^{5} - (\frac{1}{3} \int_{1}^{5} \frac{w \cdot w^{2}}{w^{1}} d w)$

Step 4.2.2.2

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

$\frac{\ln (w) w^{3}}{3}]_{1}^{5} - (\frac{1}{3} \int_{1}^{5} \frac{w \cdot w^{2}}{w \cdot 1} d w)$

Step 4.2.2.3

Cancel the common factor.

$\frac{\ln (w) w^{3}}{3}]_{1}^{5} - (\frac{1}{3} \int_{1}^{5} \frac{w \cdot w^{2}}{w \cdot 1} d w)$

Step 4.2.2.4

Rewrite the expression.

$\frac{\ln (w) w^{3}}{3}]_{1}^{5} - (\frac{1}{3} \int_{1}^{5} \frac{w^{2}}{1} d w)$

Step 4.2.2.5

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

$\frac{\ln (w) w^{3}}{3}]_{1}^{5} - (\frac{1}{3} \int_{1}^{5} w^{2} d w)$

$\frac{\ln (w) w^{3}}{3}]_{1}^{5} - \frac{1}{3} \int_{1}^{5} w^{2} d w$

Step 5

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

$\frac{\ln (w) w^{3}}{3}]_{1}^{5} - \frac{1}{3} \frac{1}{3} w^{3}]_{1}^{5}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{\ln (w) w^{3}}{3}$ at $5$ and at $1$ .

$(\frac{\ln (5) \cdot 5^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{1}{3} \frac{1}{3} w^{3}]_{1}^{5}$

Step 6.2

Evaluate $\frac{1}{3} w^{3}$ at $5$ and at $1$ .

$(\frac{\ln (5) \cdot 5^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{1}{3} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} \cdot 1^{3})$

Step 6.3

Simplify.

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Step 6.3.1

Raise $5$ to the power of $3$ .

$\frac{\ln (5) \cdot 125}{3} - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{1}{3} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} \cdot 1^{3})$

Step 6.3.2

Move $125$ to the left of $\ln (5)$ .

$\frac{125 \cdot \ln (5)}{3} - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{1}{3} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} \cdot 1^{3})$

Step 6.3.3

One to any power is one.

$\frac{125 \ln (5)}{3} - \frac{\ln (1) \cdot 1}{3} - \frac{1}{3} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} \cdot 1^{3})$

Step 6.3.4

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

$\frac{125 \ln (5)}{3} - \frac{\ln (1)}{3} - \frac{1}{3} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} \cdot 1^{3})$

Step 6.3.5

Raise $5$ to the power of $3$ .

$\frac{125 \ln (5)}{3} - \frac{\ln (1)}{3} - \frac{1}{3} (\frac{1}{3} \cdot 125 - \frac{1}{3} \cdot 1^{3})$

Step 6.3.6

Combine $\frac{1}{3}$ and $125$ .

$\frac{125 \ln (5)}{3} - \frac{\ln (1)}{3} - \frac{1}{3} (\frac{125}{3} - \frac{1}{3} \cdot 1^{3})$

Step 6.3.7

One to any power is one.

$\frac{125 \ln (5)}{3} - \frac{\ln (1)}{3} - \frac{1}{3} (\frac{125}{3} - \frac{1}{3} \cdot 1)$

Step 6.3.8

Multiply $- 1$ by $1$ .

$\frac{125 \ln (5)}{3} - \frac{\ln (1)}{3} - \frac{1}{3} (\frac{125}{3} - \frac{1}{3})$

Step 6.3.9

Combine the numerators over the common denominator.

$\frac{125 \ln (5)}{3} - \frac{\ln (1)}{3} - \frac{1}{3} \cdot \frac{125 - 1}{3}$

Step 6.3.10

Subtract $1$ from $125$ .

$\frac{125 \ln (5)}{3} - \frac{\ln (1)}{3} - \frac{1}{3} \cdot \frac{124}{3}$

Step 6.3.11

Multiply $\frac{124}{3}$ by $\frac{1}{3}$ .

$\frac{125 \ln (5)}{3} - \frac{\ln (1)}{3} - \frac{124}{3 \cdot 3}$

Step 6.3.12

Multiply $3$ by $3$ .

$\frac{125 \ln (5)}{3} - \frac{\ln (1)}{3} - \frac{124}{9}$

Step 7

Simplify.

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Step 7.1

Combine the numerators over the common denominator.

$\frac{125 \ln (5) - \ln (1)}{3} + \frac{- 124}{9}$

Step 7.2

Simplify each term.

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Step 7.2.1

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

$\frac{125 \ln (5) - 0}{3} + \frac{- 124}{9}$

Step 7.2.2

Multiply $- 1$ by $0$ .

$\frac{125 \ln (5) + 0}{3} + \frac{- 124}{9}$

Step 7.3

Add $125 \ln (5)$ and $0$ .

$\frac{125 \ln (5)}{3} + \frac{- 124}{9}$

Step 7.4

Move the negative in front of the fraction.

$\frac{125 \ln (5)}{3} - \frac{124}{9}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{125 \ln (5)}{3} - \frac{124}{9}$

Decimal Form:

$53.28213524 \dots$