Calculus Examples

$\int_{1}^{e} \ln (15 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (15 x)$ and $d v = 1$ .

$\ln (15 x) x]_{1}^{e} - \int_{1}^{e} x \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

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

$\ln (15 x) x]_{1}^{e} - \int_{1}^{e} \frac{x}{x} d x$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\ln (15 x) x]_{1}^{e} - \int_{1}^{e} \frac{x}{x} d x$

Step 2.2.2

Rewrite the expression.

$\ln (15 x) x]_{1}^{e} - \int_{1}^{e} d x$

Step 3

Apply the constant rule.

$\ln (15 x) x]_{1}^{e} - (x]_{1}^{e})$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\ln (15 x) x$ at $e$ and at $1$ .

$(\ln (15 e) e) - \ln (15 \cdot 1) \cdot 1 - (x]_{1}^{e})$

Step 4.2

Evaluate $x$ at $e$ and at $1$ .

$(\ln (15 e) e) - \ln (15 \cdot 1) \cdot 1 - ((e) - 1)$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $15$ by $1$ .

$\ln (15 e) e - \ln (15) \cdot 1 - ((e) - 1)$

Step 4.3.2

Multiply $- 1$ by $1$ .

$\ln (15 e) e - \ln (15) - (e - 1)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$\ln (15 e) e - \ln (15) - e - - 1$

Step 5.2

Multiply $- 1$ by $- 1$ .

$\ln (15 e) e - \ln (15) - e + 1$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (15 e) e - \ln (15) - e + 1$

Decimal Form:

$5.65319345 \dots$