Calculus Examples

$\int \ln (\sqrt{x}) d x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 2

Rewrite $\ln (x^{\frac{1}{2}})$ as $\frac{1}{2} \ln (x)$ .

$\int \frac{1}{2} \ln (x) d x$

Step 3

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

$\frac{1}{2} \int \ln (x) d x$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Cancel the common factor of $x$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 6

Apply the constant rule.

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

Step 7

Simplify.

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