Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = 1$ .

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

$\ln (x) x]_{1}^{5} - \int_{1}^{5} d x$

Step 3

Apply the constant rule.

$\ln (x) x]_{1}^{5} - (x]_{1}^{5})$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\ln (x) x$ at $5$ and at $1$ .

$(\ln (5) \cdot 5) - \ln (1) \cdot 1 - (x]_{1}^{5})$

Step 4.2

Evaluate $x$ at $5$ and at $1$ .

$(\ln (5) \cdot 5) - \ln (1) \cdot 1 - ((5) - 1)$

Step 4.3

Simplify.

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Step 4.3.1

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

$5 \cdot \ln (5) - \ln (1) \cdot 1 - ((5) - 1)$

Step 4.3.2

Multiply $- 1$ by $1$ .

$5 \ln (5) - \ln (1) - ((5) - 1)$

Step 4.3.3

Subtract $1$ from $5$ .

$5 \ln (5) - \ln (1) - 1 \cdot 4$

Step 4.3.4

Multiply $- 1$ by $4$ .

$5 \ln (5) - \ln (1) - 4$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

$5 \ln (5) - 0 - 4$

Step 5.1.2

Multiply $- 1$ by $0$ .

$5 \ln (5) + 0 - 4$

Step 5.2

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

$5 \ln (5) - 4$

Step 6

The result can be shown in multiple forms.

Exact Form:

$5 \ln (5) - 4$

Decimal Form:

$4.04718956 \dots$