Calculus Examples

$\int_{0}^{1} \ln (x^{2}) d x$

Step 1

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

$\int_{0}^{1} 2 \ln (x) d x$

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{0}^{1} \ln (x) d x$

Step 3

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

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

Step 4.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 5

Apply the constant rule.

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

Step 6

Evaluate $\ln (x) x$ at $1$ and at $0$ .

$2 ((\ln (1) \cdot 1) - \ln (0) x - (x]_{0}^{1}))$

Step 7

The natural logarithm of zero is undefined.

Undefined