Calculus Examples

$\int_{1}^{2} \ln (14 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (14 x)$ and $d v = 1$ .

$\ln (14 x) x]_{1}^{2} - \int_{1}^{2} x \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

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

$\ln (14 x) x]_{1}^{2} - \int_{1}^{2} \frac{x}{x} d x$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\ln (14 x) x]_{1}^{2} - \int_{1}^{2} \frac{x}{x} d x$

Step 2.2.2

Rewrite the expression.

$\ln (14 x) x]_{1}^{2} - \int_{1}^{2} d x$

Step 3

Apply the constant rule.

$\ln (14 x) x]_{1}^{2} - (x]_{1}^{2})$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\ln (14 x) x$ at $2$ and at $1$ .

$(\ln (14 \cdot 2) \cdot 2) - \ln (14 \cdot 1) \cdot 1 - (x]_{1}^{2})$

Step 4.2

Evaluate $x$ at $2$ and at $1$ .

$(\ln (14 \cdot 2) \cdot 2) - \ln (14 \cdot 1) \cdot 1 - ((2) - 1)$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $14$ by $2$ .

$\ln (28) \cdot 2 - \ln (14 \cdot 1) \cdot 1 - ((2) - 1)$

Step 4.3.2

Move $2$ to the left of $\ln (28)$ .

$2 \cdot \ln (28) - \ln (14 \cdot 1) \cdot 1 - ((2) - 1)$

Step 4.3.3

Multiply $14$ by $1$ .

$2 \ln (28) - \ln (14) \cdot 1 - ((2) - 1)$

Step 4.3.4

Multiply $- 1$ by $1$ .

$2 \ln (28) - \ln (14) - ((2) - 1)$

Step 4.3.5

Subtract $1$ from $2$ .

$2 \ln (28) - \ln (14) - 1 \cdot 1$

Step 4.3.6

Multiply $- 1$ by $1$ .

$2 \ln (28) - \ln (14) - 1$

Step 5

The result can be shown in multiple forms.

Exact Form:

$2 \ln (28) - \ln (14) - 1$

Decimal Form:

$3.02535169 \dots$