Calculus Examples

$\int_{2}^{3} \frac{2 + u^{2}}{u^{3}} d u$

Step 1

Apply basic rules of exponents.

Tap for more steps...

Step 1.1

Move $u^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int_{2}^{3} (2 + u^{2}) {(u^{3})}^{- 1} d u$

Step 1.2

Multiply the exponents in ${(u^{3})}^{- 1}$ .

Tap for more steps...

Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int_{2}^{3} (2 + u^{2}) u^{3 \cdot - 1} d u$

Step 1.2.2

Multiply $3$ by $- 1$ .

$\int_{2}^{3} (2 + u^{2}) u^{- 3} d u$

Step 2

Multiply $(2 + u^{2}) u^{- 3}$ .

$\int_{2}^{3} 2 u^{- 3} + u^{2} u^{- 3} d u$

Step 3

Multiply $u^{2}$ by $u^{- 3}$ by adding the exponents.

Tap for more steps...

Step 3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{2}^{3} 2 u^{- 3} + u^{2 - 3} d u$

Step 3.2

Subtract $3$ from $2$ .

$\int_{2}^{3} 2 u^{- 3} + u^{- 1} d u$

Step 4

Split the single integral into multiple integrals.

$\int_{2}^{3} 2 u^{- 3} d u + \int_{2}^{3} u^{- 1} d u$

Step 5

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

$2 \int_{2}^{3} u^{- 3} d u + \int_{2}^{3} u^{- 1} d u$

Step 6

By the Power Rule, the integral of $u^{- 3}$ with respect to $u$ is $- \frac{1}{2} u^{- 2}$ .

$2 (- \frac{1}{2} u^{- 2}]_{2}^{3}) + \int_{2}^{3} u^{- 1} d u$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Combine $u^{- 2}$ and $\frac{1}{2}$ .

$2 (- \frac{u^{- 2}}{2}]_{2}^{3}) + \int_{2}^{3} u^{- 1} d u$

Step 7.2

Move $u^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 (- \frac{1}{2 u^{2}}]_{2}^{3}) + \int_{2}^{3} u^{- 1} d u$

Step 8

The integral of $u^{- 1}$ with respect to $u$ is $\ln (| u |)$ .

$2 (- \frac{1}{2 u^{2}}]_{2}^{3}) + \ln (| u |)]_{2}^{3}$

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Substitute and simplify.

Tap for more steps...

Step 9.1.1

Evaluate $- \frac{1}{2 u^{2}}$ at $3$ and at $2$ .

$2 ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 2^{2}}) + \ln (| u |)]_{2}^{3}$

Step 9.1.2

Evaluate $\ln (| u |)$ at $3$ and at $2$ .

$2 ((- \frac{1}{2 \cdot 3^{2}}) + \frac{1}{2 \cdot 2^{2}}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3

Simplify.

Tap for more steps...

Step 9.1.3.1

Raise $3$ to the power of $2$ .

$2 (- \frac{1}{2 \cdot 9} + \frac{1}{2 \cdot 2^{2}}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.2

Multiply $2$ by $9$ .

$2 (- \frac{1}{18} + \frac{1}{2 \cdot 2^{2}}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.3

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

$2 (- \frac{1}{18} + \frac{1}{2 \cdot 4}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.4

Multiply $2$ by $4$ .

$2 (- \frac{1}{18} + \frac{1}{8}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.5

To write $- \frac{1}{18}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 (- \frac{1}{18} \cdot \frac{4}{4} + \frac{1}{8}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.6

To write $\frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$2 (- \frac{1}{18} \cdot \frac{4}{4} + \frac{1}{8} \cdot \frac{9}{9}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.7

Write each expression with a common denominator of $72$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 9.1.3.7.1

Multiply $\frac{1}{18}$ by $\frac{4}{4}$ .

$2 (- \frac{4}{18 \cdot 4} + \frac{1}{8} \cdot \frac{9}{9}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.7.2

Multiply $18$ by $4$ .

$2 (- \frac{4}{72} + \frac{1}{8} \cdot \frac{9}{9}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.7.3

Multiply $\frac{1}{8}$ by $\frac{9}{9}$ .

$2 (- \frac{4}{72} + \frac{9}{8 \cdot 9}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.7.4

Multiply $8$ by $9$ .

$2 (- \frac{4}{72} + \frac{9}{72}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.8

Combine the numerators over the common denominator.

$2 \frac{- 4 + 9}{72} + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.9

Add $- 4$ and $9$ .

$2 (\frac{5}{72}) + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.10

Combine $2$ and $\frac{5}{72}$ .

$\frac{2 \cdot 5}{72} + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.11

Multiply $2$ by $5$ .

$\frac{10}{72} + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.12

Cancel the common factor of $10$ and $72$ .

Tap for more steps...

Step 9.1.3.12.1

Factor $2$ out of $10$ .

$\frac{2 (5)}{72} + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.12.2

Cancel the common factors.

Tap for more steps...

Step 9.1.3.12.2.1

Factor $2$ out of $72$ .

$\frac{2 \cdot 5}{2 \cdot 36} + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.12.2.2

Cancel the common factor.

$\frac{2 \cdot 5}{2 \cdot 36} + (\ln (| 3 |)) - \ln (| 2 |)$

Step 9.1.3.12.2.3

Rewrite the expression.

$\frac{5}{36} + (\ln (| 3 |)) - \ln (| 2 |)$

$\frac{5}{36} + \ln (| 3 |) - \ln (| 2 |)$

Step 9.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{5}{36} + \ln (\frac{| 3 |}{| 2 |})$

Step 9.3

Simplify.

Tap for more steps...

Step 9.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{5}{36} + \ln (\frac{3}{| 2 |})$

Step 9.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{5}{36} + \ln (\frac{3}{2})$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{36} + \ln (\frac{3}{2})$

Decimal Form:

$0.54435399 \dots$

Step 11