Calculus Examples

$\int x^{- 6} \ln (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{- 6}$ .

$\ln (x) (- \frac{1}{5 x^{5}}) - \int - \frac{1}{5 x^{5}} \cdot \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

Combine $\ln (x)$ and $\frac{1}{5 x^{5}}$ .

$- \frac{\ln (x)}{5 x^{5}} - \int - \frac{1}{5 x^{5}} \cdot \frac{1}{x} d x$

Step 2.2

Multiply $\frac{1}{x}$ by $\frac{1}{5 x^{5}}$ .

$- \frac{\ln (x)}{5 x^{5}} - \int - \frac{1}{x (5 x^{5})} d x$

Step 2.3

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

$- \frac{\ln (x)}{5 x^{5}} - \int - \frac{1}{5 (x^{1} x^{5})} d x$

Step 2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{\ln (x)}{5 x^{5}} - \int - \frac{1}{5 x^{1 + 5}} d x$

Step 2.5

Add $1$ and $5$ .

$- \frac{\ln (x)}{5 x^{5}} - \int - \frac{1}{5 x^{6}} d x$

Step 3

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

$- \frac{\ln (x)}{5 x^{5}} - - \int \frac{1}{5 x^{6}} d x$

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$- \frac{\ln (x)}{5 x^{5}} + 1 \int \frac{1}{5 x^{6}} d x$

Step 4.2

Multiply $\int \frac{1}{5 x^{6}} d x$ by $1$ .

$- \frac{\ln (x)}{5 x^{5}} + \int \frac{1}{5 x^{6}} d x$

Step 5

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$- \frac{\ln (x)}{5 x^{5}} + \frac{1}{5} \int \frac{1}{x^{6}} d x$

Step 6

Apply basic rules of exponents.

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Step 6.1

Move $x^{6}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{\ln (x)}{5 x^{5}} + \frac{1}{5} \int {(x^{6})}^{- 1} d x$

Step 6.2

Multiply the exponents in ${(x^{6})}^{- 1}$ .

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Step 6.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{\ln (x)}{5 x^{5}} + \frac{1}{5} \int x^{6 \cdot - 1} d x$

Step 6.2.2

Multiply $6$ by $- 1$ .

$- \frac{\ln (x)}{5 x^{5}} + \frac{1}{5} \int x^{- 6} d x$

Step 7

By the Power Rule, the integral of $x^{- 6}$ with respect to $x$ is $- \frac{1}{5} x^{- 5}$ .

$- \frac{\ln (x)}{5 x^{5}} + \frac{1}{5} (- \frac{1}{5} x^{- 5} + C)$

Step 8

Simplify the answer.

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Step 8.1

Rewrite $- \frac{\ln (x)}{5 x^{5}} + \frac{1}{5} (- \frac{1}{5} x^{- 5} + C)$ as $- \frac{\ln (x)}{5 x^{5}} + \frac{1}{5} (- \frac{1}{5} \cdot \frac{1}{x^{5}}) + C$ .

$- \frac{\ln (x)}{5 x^{5}} + \frac{1}{5} (- \frac{1}{5} \cdot \frac{1}{x^{5}}) + C$

Step 8.2

Simplify.

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Step 8.2.1

Multiply $\frac{1}{x^{5}}$ by $\frac{1}{5}$ .

$- \frac{\ln (x)}{5 x^{5}} + \frac{1}{5} (- \frac{1}{x^{5} \cdot 5}) + C$

Step 8.2.2

Move $5$ to the left of $x^{5}$ .

$- \frac{\ln (x)}{5 x^{5}} + \frac{1}{5} (- \frac{1}{5 \cdot x^{5}}) + C$

Step 8.2.3

Multiply $\frac{1}{5}$ by $\frac{1}{5 x^{5}}$ .

$- \frac{\ln (x)}{5 x^{5}} - \frac{1}{5 (5 x^{5})} + C$

Step 8.2.4

Multiply $5$ by $5$ .

$- \frac{\ln (x)}{5 x^{5}} - \frac{1}{25 x^{5}} + C$