Calculus Examples

$\int (2 x - 1) \ln (7 x) d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int 2 x \ln (7 x) - 1 \ln (7 x) d x$

Step 1.2

Simplify $2 \ln (7 x)$ by moving $2$ inside the logarithm.

$\int x \ln ({(7 x)}^{2}) - 1 \ln (7 x) d x$

Step 1.3

Rewrite $- 1 \ln (7 x)$ as $- \ln (7 x)$ .

$\int x \ln ({(7 x)}^{2}) - \ln (7 x) d x$

Step 1.4

Simplify each term.

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Step 1.4.1

Apply the product rule to $7 x$ .

$\int x \ln (7^{2} x^{2}) - \ln (7 x) d x$

Step 1.4.2

Raise $7$ to the power of $2$ .

$\int x \ln (49 x^{2}) - \ln (7 x) d x$

Step 2

Split the single integral into multiple integrals.

$\int x \ln (49 x^{2}) d x + \int - \ln (7 x) d x$

Step 3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (49 x^{2})$ and $d v = x$ .

$\ln (49 x^{2}) (\frac{1}{2} x^{2}) - \int \frac{1}{2} x^{2} \frac{2}{x} d x + \int - \ln (7 x) d x$

Step 4

Simplify.

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Step 4.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\ln (49 x^{2}) \frac{x^{2}}{2} - \int \frac{1}{2} x^{2} \frac{2}{x} d x + \int - \ln (7 x) d x$

Step 4.2

Combine $\ln (49 x^{2})$ and $\frac{x^{2}}{2}$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} - \int \frac{1}{2} x^{2} \frac{2}{x} d x + \int - \ln (7 x) d x$

Step 5

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{1}{2} \int x^{2} \frac{2}{x} d x) + \int - \ln (7 x) d x$

Step 6

Simplify.

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Step 6.1

Combine $x^{2}$ and $\frac{2}{x}$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{1}{2} \int \frac{x^{2} \cdot 2}{x} d x) + \int - \ln (7 x) d x$

Step 6.2

Move $2$ to the left of $x^{2}$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{1}{2} \int \frac{2 \cdot x^{2}}{x} d x) + \int - \ln (7 x) d x$

Step 6.3

Cancel the common factor of $x^{2}$ and $x$ .

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Step 6.3.1

Factor $x$ out of $2 x^{2}$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{1}{2} \int \frac{x (2 x)}{x} d x) + \int - \ln (7 x) d x$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

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

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{1}{2} \int \frac{x (2 x)}{x^{1}} d x) + \int - \ln (7 x) d x$

Step 6.3.2.2

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

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{1}{2} \int \frac{x (2 x)}{x \cdot 1} d x) + \int - \ln (7 x) d x$

Step 6.3.2.3

Cancel the common factor.

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{1}{2} \int \frac{x (2 x)}{x \cdot 1} d x) + \int - \ln (7 x) d x$

Step 6.3.2.4

Rewrite the expression.

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{1}{2} \int \frac{2 x}{1} d x) + \int - \ln (7 x) d x$

Step 6.3.2.5

Divide $2 x$ by $1$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{1}{2} \int 2 x d x) + \int - \ln (7 x) d x$

$\frac{\ln (49 x^{2}) x^{2}}{2} - \frac{1}{2} \int 2 x d x + \int - \ln (7 x) d x$

Step 7

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\frac{\ln (49 x^{2}) x^{2}}{2} - \frac{1}{2} (2 \int x d x) + \int - \ln (7 x) d x$

Step 8

Simplify.

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Step 8.1

Multiply $2$ by $- 1$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} - 2 (\frac{1}{2}) \int x d x + \int - \ln (7 x) d x$

Step 8.2

Combine $- 2$ and $\frac{1}{2}$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} + \frac{- 2}{2} \int x d x + \int - \ln (7 x) d x$

Step 8.3

Cancel the common factor of $- 2$ and $2$ .

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Step 8.3.1

Factor $2$ out of $- 2$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} + \frac{2 \cdot - 1}{2} \int x d x + \int - \ln (7 x) d x$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $2$ out of $2$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} + \frac{2 \cdot - 1}{2 (1)} \int x d x + \int - \ln (7 x) d x$

Step 8.3.2.2

Cancel the common factor.

$\frac{\ln (49 x^{2}) x^{2}}{2} + \frac{2 \cdot - 1}{2 \cdot 1} \int x d x + \int - \ln (7 x) d x$

Step 8.3.2.3

Rewrite the expression.

$\frac{\ln (49 x^{2}) x^{2}}{2} + \frac{- 1}{1} \int x d x + \int - \ln (7 x) d x$

Step 8.3.2.4

Divide $- 1$ by $1$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} - \int x d x + \int - \ln (7 x) d x$

Step 9

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

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{1}{2} x^{2} + C) + \int - \ln (7 x) d x$

Step 10

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{x^{2}}{2} + C) + \int - \ln (7 x) d x$

Step 11

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{x^{2}}{2} + C) - \int \ln (7 x) d x$

Step 12

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (7 x)$ and $d v = 1$ .

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{x^{2}}{2} + C) - (\ln (7 x) x - \int x \frac{1}{x} d x)$

Step 13

Simplify.

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Step 13.1

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

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{x^{2}}{2} + C) - (\ln (7 x) x - \int \frac{x}{x} d x)$

Step 13.2

Cancel the common factor of $x$ .

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Step 13.2.1

Cancel the common factor.

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{x^{2}}{2} + C) - (\ln (7 x) x - \int \frac{x}{x} d x)$

Step 13.2.2

Rewrite the expression.

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{x^{2}}{2} + C) - (\ln (7 x) x - \int d x)$

Step 14

Apply the constant rule.

$\frac{\ln (49 x^{2}) x^{2}}{2} - (\frac{x^{2}}{2} + C) - (\ln (7 x) x - (x + C))$

Step 15

Simplify.

$\frac{\ln (49 x^{2}) x^{2}}{2} - \frac{x^{2}}{2} - \ln (7 x) x + x + C$

Step 16

Reorder terms.

$\frac{1}{2} \ln (49 x^{2}) x^{2} - \frac{1}{2} x^{2} - \ln (7 x) x + x + C$