Calculus Examples

$\int_{1}^{7} \ln (20 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (20 x)$ and $d v = 1$ .

$\ln (20 x) x]_{1}^{7} - \int_{1}^{7} x \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

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

$\ln (20 x) x]_{1}^{7} - \int_{1}^{7} \frac{x}{x} d x$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\ln (20 x) x]_{1}^{7} - \int_{1}^{7} \frac{x}{x} d x$

Step 2.2.2

Rewrite the expression.

$\ln (20 x) x]_{1}^{7} - \int_{1}^{7} d x$

Step 3

Apply the constant rule.

$\ln (20 x) x]_{1}^{7} - (x]_{1}^{7})$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\ln (20 x) x$ at $7$ and at $1$ .

$(\ln (20 \cdot 7) \cdot 7) - \ln (20 \cdot 1) \cdot 1 - (x]_{1}^{7})$

Step 4.2

Evaluate $x$ at $7$ and at $1$ .

$(\ln (20 \cdot 7) \cdot 7) - \ln (20 \cdot 1) \cdot 1 - ((7) - 1)$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $20$ by $7$ .

$\ln (140) \cdot 7 - \ln (20 \cdot 1) \cdot 1 - ((7) - 1)$

Step 4.3.2

Move $7$ to the left of $\ln (140)$ .

$7 \cdot \ln (140) - \ln (20 \cdot 1) \cdot 1 - ((7) - 1)$

Step 4.3.3

Multiply $20$ by $1$ .

$7 \ln (140) - \ln (20) \cdot 1 - ((7) - 1)$

Step 4.3.4

Multiply $- 1$ by $1$ .

$7 \ln (140) - \ln (20) - ((7) - 1)$

Step 4.3.5

Subtract $1$ from $7$ .

$7 \ln (140) - \ln (20) - 1 \cdot 6$

Step 4.3.6

Multiply $- 1$ by $6$ .

$7 \ln (140) - \ln (20) - 6$

Step 5

The result can be shown in multiple forms.

Exact Form:

$7 \ln (140) - \ln (20) - 6$

Decimal Form:

$25.59576468 \dots$