Calculus Examples

$\int_{0}^{2} \ln (x) d x$

Step 1

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

$\ln (x) x]_{0}^{2} - \int_{0}^{2} x \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

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

$\ln (x) x]_{0}^{2} - \int_{0}^{2} \frac{x}{x} d x$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\ln (x) x]_{0}^{2} - \int_{0}^{2} \frac{x}{x} d x$

Step 2.2.2

Rewrite the expression.

$\ln (x) x]_{0}^{2} - \int_{0}^{2} d x$

Step 3

Apply the constant rule.

$\ln (x) x]_{0}^{2} - (x]_{0}^{2})$

Step 4

Evaluate $\ln (x) x$ at $2$ and at $0$ .

$(\ln (2) \cdot 2) - \ln (0) x - (x]_{0}^{2})$

Step 5

The natural logarithm of zero is undefined.

Undefined