Calculus Examples

$\int x \log_{5} (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \log_{5} (x)$ and $d v = x$ .

$\log_{5} (x) (\frac{1}{2} x^{2}) - \int \frac{1}{2} x^{2} \frac{1}{x \ln (5)} d x$

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

$\log_{5} (x) \frac{x^{2}}{2} - \int \frac{1}{2} x^{2} \frac{1}{x \ln (5)} d x$

Step 2.2

Combine $\log_{5} (x)$ and $\frac{x^{2}}{2}$ .

$\frac{\log_{5} (x) x^{2}}{2} - \int \frac{1}{2} x^{2} \frac{1}{x \ln (5)} d x$

Step 3

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

$\frac{\log_{5} (x) x^{2}}{2} - (\frac{1}{2} \int x^{2} \frac{1}{x \ln (5)} d x)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Combine $x^{2}$ and $\frac{1}{x \ln (5)}$ .

$\frac{\log_{5} (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x^{2}}{x \ln (5)} d x)$

Step 4.2

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

Tap for more steps...

Step 4.2.1

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

$\frac{\log_{5} (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x \cdot x}{x \ln (5)} d x)$

Step 4.2.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.1

Factor $x$ out of $x \ln (5)$ .

$\frac{\log_{5} (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x \cdot x}{x (\ln (5))} d x)$

Step 4.2.2.2

Cancel the common factor.

$\frac{\log_{5} (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x \cdot x}{x \ln (5)} d x)$

Step 4.2.2.3

Rewrite the expression.

$\frac{\log_{5} (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x}{\ln (5)} d x)$

$\frac{\log_{5} (x) x^{2}}{2} - \frac{1}{2} \int \frac{x}{\ln (5)} d x$

Step 5

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

$\frac{\log_{5} (x) x^{2}}{2} - \frac{1}{2} (\frac{1}{\ln (5)} \int x d x)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply $\frac{1}{\ln (5)}$ by $\frac{1}{2}$ .

$\frac{\log_{5} (x) x^{2}}{2} - \frac{1}{\ln (5) \cdot 2} \int x d x$

Step 6.2

Move $2$ to the left of $\ln (5)$ .

$\frac{\log_{5} (x) x^{2}}{2} - \frac{1}{2 \ln (5)} \int x d x$

Step 7

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

$\frac{\log_{5} (x) x^{2}}{2} - \frac{1}{2 \ln (5)} (\frac{1}{2} x^{2} + C)$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

Rewrite $\frac{\log_{5} (x) x^{2}}{2} - \frac{1}{2 \ln (5)} (\frac{1}{2} x^{2} + C)$ as $\frac{1}{2} \log_{5} (x) x^{2} - \frac{1}{2 \ln (5)} \cdot \frac{1}{2} x^{2} + C$ .

$\frac{1}{2} \log_{5} (x) x^{2} - \frac{1}{2 \ln (5)} \cdot \frac{1}{2} x^{2} + C$

Step 8.2

Simplify.

Tap for more steps...

Step 8.2.1

Combine $\frac{1}{2}$ and $\log_{5} (x)$ .

$\frac{\log_{5} (x)}{2} x^{2} - \frac{1}{2 \ln (5)} \cdot \frac{1}{2} x^{2} + C$

Step 8.2.2

Combine $\frac{\log_{5} (x)}{2}$ and $x^{2}$ .

$\frac{\log_{5} (x) x^{2}}{2} - \frac{1}{2 \ln (5)} \cdot \frac{1}{2} x^{2} + C$

Step 8.2.3

Multiply $\frac{1}{2}$ by $\frac{1}{2 \ln (5)}$ .

$\frac{\log_{5} (x) x^{2}}{2} - \frac{1}{2 (2 \ln (5))} x^{2} + C$

Step 8.2.4

Multiply $2$ by $2$ .

$\frac{\log_{5} (x) x^{2}}{2} - \frac{1}{4 \ln (5)} x^{2} + C$

Step 8.3

Combine $x^{2}$ and $\frac{1}{4 \ln (5)}$ .

$\frac{\log_{5} (x) x^{2}}{2} - \frac{x^{2}}{4 \ln (5)} + C$

Step 8.4

Reorder terms.

$\frac{1}{2} \log_{5} (x) x^{2} - \frac{1}{4 \ln (5)} x^{2} + C$