Calculus Examples

$\int_{1}^{2} x^{5} \ln (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{5}$ .

$\ln (x) (\frac{1}{6} x^{6})]_{1}^{2} - \int_{1}^{2} \frac{1}{6} x^{6} \frac{1}{x} d x$

Step 2

Simplify.

Tap for more steps...

Combine $\frac{1}{6}$ and $x^{6}$ .

$\ln (x) \frac{x^{6}}{6}]_{1}^{2} - \int_{1}^{2} \frac{1}{6} x^{6} \frac{1}{x} d x$

Combine $\ln (x)$ and $\frac{x^{6}}{6}$ .

$\frac{\ln (x) x^{6}}{6}]_{1}^{2} - \int_{1}^{2} \frac{1}{6} x^{6} \frac{1}{x} d x$

Step 3

Since $\frac{1}{6}$ is constant with respect to $x$ , move $\frac{1}{6}$ out of the integral.

$\frac{\ln (x) x^{6}}{6}]_{1}^{2} - (\frac{1}{6} \int_{1}^{2} x^{6} \frac{1}{x} d x)$

Step 4

Simplify.

Tap for more steps...

Combine $x^{6}$ and $\frac{1}{x}$ .

$\frac{\ln (x) x^{6}}{6}]_{1}^{2} - (\frac{1}{6} \int_{1}^{2} \frac{x^{6}}{x} d x)$

Cancel the common factor of $x^{6}$ and $x$ .

Tap for more steps...

Factor $x$ out of $x^{6}$ .

$\frac{\ln (x) x^{6}}{6}]_{1}^{2} - (\frac{1}{6} \int_{1}^{2} \frac{x \cdot x^{5}}{x} d x)$

Cancel the common factors.

Tap for more steps...

Raise $x$ to the power of $1$ .

$\frac{\ln (x) x^{6}}{6}]_{1}^{2} - (\frac{1}{6} \int_{1}^{2} \frac{x \cdot x^{5}}{x^{1}} d x)$

Factor $x$ out of $x^{1}$ .

$\frac{\ln (x) x^{6}}{6}]_{1}^{2} - (\frac{1}{6} \int_{1}^{2} \frac{x \cdot x^{5}}{x \cdot 1} d x)$

Cancel the common factor.

$\frac{\ln (x) x^{6}}{6}]_{1}^{2} - (\frac{1}{6} \int_{1}^{2} \frac{x \cdot x^{5}}{x \cdot 1} d x)$

Rewrite the expression.

$\frac{\ln (x) x^{6}}{6}]_{1}^{2} - (\frac{1}{6} \int_{1}^{2} \frac{x^{5}}{1} d x)$

Divide $x^{5}$ by $1$ .

$\frac{\ln (x) x^{6}}{6}]_{1}^{2} - (\frac{1}{6} \int_{1}^{2} x^{5} d x)$

$\frac{\ln (x) x^{6}}{6}]_{1}^{2} - \frac{1}{6} \int_{1}^{2} x^{5} d x$

Step 5

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$\frac{\ln (x) x^{6}}{6}]_{1}^{2} - \frac{1}{6} \frac{1}{6} x^{6}]_{1}^{2}$

Step 6

Substitute and simplify.

Tap for more steps...

Evaluate $\frac{\ln (x) x^{6}}{6}$ at $2$ and at $1$ .

$(\frac{\ln (2) \cdot 2^{6}}{6}) - \frac{\ln (1) \cdot 1^{6}}{6} - \frac{1}{6} \frac{1}{6} x^{6}]_{1}^{2}$

Evaluate $\frac{1}{6} x^{6}$ at $2$ and at $1$ .

$(\frac{\ln (2) \cdot 2^{6}}{6}) - \frac{\ln (1) \cdot 1^{6}}{6} - \frac{1}{6} ((\frac{1}{6} \cdot 2^{6}) - \frac{1}{6} \cdot 1^{6})$

Simplify.

Tap for more steps...

Raise $2$ to the power of $6$ .

$\frac{\ln (2) \cdot 64}{6} - \frac{\ln (1) \cdot 1^{6}}{6} - \frac{1}{6} ((\frac{1}{6} \cdot 2^{6}) - \frac{1}{6} \cdot 1^{6})$

Move $64$ to the left of $\ln (2)$ .

$\frac{64 \cdot \ln (2)}{6} - \frac{\ln (1) \cdot 1^{6}}{6} - \frac{1}{6} ((\frac{1}{6} \cdot 2^{6}) - \frac{1}{6} \cdot 1^{6})$

Cancel the common factor of $64$ and $6$ .

Tap for more steps...

Factor $2$ out of $64 \ln (2)$ .

$\frac{2 (32 \ln (2))}{6} - \frac{\ln (1) \cdot 1^{6}}{6} - \frac{1}{6} ((\frac{1}{6} \cdot 2^{6}) - \frac{1}{6} \cdot 1^{6})$

Cancel the common factors.

Tap for more steps...

Factor $2$ out of $6$ .

$\frac{2 (32 \ln (2))}{2 (3)} - \frac{\ln (1) \cdot 1^{6}}{6} - \frac{1}{6} ((\frac{1}{6} \cdot 2^{6}) - \frac{1}{6} \cdot 1^{6})$

Cancel the common factor.

$\frac{2 (32 \ln (2))}{2 \cdot 3} - \frac{\ln (1) \cdot 1^{6}}{6} - \frac{1}{6} ((\frac{1}{6} \cdot 2^{6}) - \frac{1}{6} \cdot 1^{6})$

Rewrite the expression.

$\frac{32 \ln (2)}{3} - \frac{\ln (1) \cdot 1^{6}}{6} - \frac{1}{6} ((\frac{1}{6} \cdot 2^{6}) - \frac{1}{6} \cdot 1^{6})$

One to any power is one.

$\frac{32 \ln (2)}{3} - \frac{\ln (1) \cdot 1}{6} - \frac{1}{6} ((\frac{1}{6} \cdot 2^{6}) - \frac{1}{6} \cdot 1^{6})$

Multiply $\ln (1)$ by $1$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} ((\frac{1}{6} \cdot 2^{6}) - \frac{1}{6} \cdot 1^{6})$

Raise $2$ to the power of $6$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} (\frac{1}{6} \cdot 64 - \frac{1}{6} \cdot 1^{6})$

Combine $\frac{1}{6}$ and $64$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} (\frac{64}{6} - \frac{1}{6} \cdot 1^{6})$

Cancel the common factor of $64$ and $6$ .

Tap for more steps...

Factor $2$ out of $64$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} (\frac{2 (32)}{6} - \frac{1}{6} \cdot 1^{6})$

Cancel the common factors.

Tap for more steps...

Factor $2$ out of $6$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} (\frac{2 \cdot 32}{2 \cdot 3} - \frac{1}{6} \cdot 1^{6})$

Cancel the common factor.

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} (\frac{2 \cdot 32}{2 \cdot 3} - \frac{1}{6} \cdot 1^{6})$

Rewrite the expression.

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} (\frac{32}{3} - \frac{1}{6} \cdot 1^{6})$

One to any power is one.

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} (\frac{32}{3} - \frac{1}{6} \cdot 1)$

Multiply $- 1$ by $1$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} (\frac{32}{3} - \frac{1}{6})$

To write $\frac{32}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} (\frac{32}{3} \cdot \frac{2}{2} - \frac{1}{6})$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Multiply $\frac{32}{3}$ by $\frac{2}{2}$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} (\frac{32 \cdot 2}{3 \cdot 2} - \frac{1}{6})$

Multiply $3$ by $2$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} (\frac{32 \cdot 2}{6} - \frac{1}{6})$

Combine the numerators over the common denominator.

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} \cdot \frac{32 \cdot 2 - 1}{6}$

Simplify the numerator.

Tap for more steps...

Multiply $32$ by $2$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} \cdot \frac{64 - 1}{6}$

Subtract $1$ from $64$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} \cdot \frac{63}{6}$

Cancel the common factor of $63$ and $6$ .

Tap for more steps...

Factor $3$ out of $63$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} \cdot \frac{3 (21)}{6}$

Cancel the common factors.

Tap for more steps...

Factor $3$ out of $6$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} \cdot \frac{3 \cdot 21}{3 \cdot 2}$

Cancel the common factor.

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} \cdot \frac{3 \cdot 21}{3 \cdot 2}$

Rewrite the expression.

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{1}{6} \cdot \frac{21}{2}$

Multiply $\frac{21}{2}$ by $\frac{1}{6}$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{21}{2 \cdot 6}$

Multiply $2$ by $6$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{21}{12}$

Cancel the common factor of $21$ and $12$ .

Tap for more steps...

Factor $3$ out of $21$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{3 (7)}{12}$

Cancel the common factors.

Tap for more steps...

Factor $3$ out of $12$ .

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{3 \cdot 7}{3 \cdot 4}$

Cancel the common factor.

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{3 \cdot 7}{3 \cdot 4}$

Rewrite the expression.

$\frac{32 \ln (2)}{3} - \frac{\ln (1)}{6} - \frac{7}{4}$

Step 7

Evaluate.

Tap for more steps...

Simplify each term.

Tap for more steps...

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

$\frac{32 \ln (2)}{3} - \frac{0}{6} - \frac{7}{4}$

Divide $0$ by $6$ .

$\frac{32 \ln (2)}{3} - 0 - \frac{7}{4}$

Multiply $- 1$ by $0$ .

$\frac{32 \ln (2)}{3} + 0 - \frac{7}{4}$

Add $\frac{32 \ln (2)}{3}$ and $0$ .

$\frac{32 \ln (2)}{3} - \frac{7}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{32 \ln (2)}{3} - \frac{7}{4}$

Decimal Form:

$5.64356992 \dots$