Calculus Examples

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

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

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

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

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

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

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

Simplify.

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

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

Cancel the common factor of $x$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Apply the constant rule.

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

Simplify.

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