Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

Apply the constant rule.

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

Step 4

Substitute and simplify.

Tap for more steps...

Step 4.1

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

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

Step 4.2

Evaluate $x$ at $e$ and at $1$ .

$(\ln (e) e) - \ln (1) \cdot 1 - ((e) - 1)$

Step 4.3

Multiply $- 1$ by $1$ .

$\ln (e) e - \ln (1) - (e - 1)$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

The natural logarithm of $e$ is $1$ .

$1 e - \ln (1) - (e - 1)$

Step 5.1.2

Multiply $e$ by $1$ .

$e - \ln (1) - (e - 1)$

Step 5.1.3

The natural logarithm of $1$ is $0$ .

$e - 0 - (e - 1)$

Step 5.1.4

Multiply $- 1$ by $0$ .

$e + 0 - (e - 1)$

Step 5.1.5

Apply the distributive property.

$e + 0 - e - - 1$

Step 5.1.6

Multiply $- 1$ by $- 1$ .

$e + 0 - e + 1$

Step 5.2

Add $e$ and $0$ .

$e - e + 1$

Step 5.3

Subtract $e$ from $e$ .

$0 + 1$

Step 5.4

Add $0$ and $1$ .

$1$