Calculus Examples

$\int_{2}^{3} (\frac{4}{x^{2}} + 1) d x$

Step 1

Remove parentheses.

$\int_{2}^{3} \frac{4}{x^{2}} + 1 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{2}^{3} \frac{4}{x^{2}} d x + \int_{2}^{3} d x$

Step 3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{2}^{3} \frac{1}{x^{2}} d x + \int_{2}^{3} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$4 \int_{2}^{3} {(x^{2})}^{- 1} d x + \int_{2}^{3} d x$

Step 4.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.2.1

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

$4 \int_{2}^{3} x^{2 \cdot - 1} d x + \int_{2}^{3} d x$

Step 4.2.2

Multiply $2$ by $- 1$ .

$4 \int_{2}^{3} x^{- 2} d x + \int_{2}^{3} d x$

Step 5

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$4 (- x^{- 1}]_{2}^{3}) + \int_{2}^{3} d x$

Step 6

Apply the constant rule.

$4 (- x^{- 1}]_{2}^{3}) + x]_{2}^{3}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $- x^{- 1}$ at $3$ and at $2$ .

$4 ((- 3^{- 1}) + 2^{- 1}) + x]_{2}^{3}$

Step 7.2

Evaluate $x$ at $3$ and at $2$ .

$4 (- 3^{- 1} + 2^{- 1}) + 3 - 2$

Step 7.3

Simplify.

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Step 7.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 (- \frac{1}{3} + 2^{- 1}) + 3 - 2$

Step 7.3.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 (- \frac{1}{3} + \frac{1}{2}) + 3 - 2$

Step 7.3.3

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7.3.4

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 7.3.5

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

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Step 7.3.5.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

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

Step 7.3.5.2

Multiply $3$ by $2$ .

$4 (- \frac{2}{6} + \frac{1}{2} \cdot \frac{3}{3}) + 3 - 2$

Step 7.3.5.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$4 (- \frac{2}{6} + \frac{3}{2 \cdot 3}) + 3 - 2$

Step 7.3.5.4

Multiply $2$ by $3$ .

$4 (- \frac{2}{6} + \frac{3}{6}) + 3 - 2$

Step 7.3.6

Combine the numerators over the common denominator.

$4 \frac{- 2 + 3}{6} + 3 - 2$

Step 7.3.7

Add $- 2$ and $3$ .

$4 (\frac{1}{6}) + 3 - 2$

Step 7.3.8

Combine $4$ and $\frac{1}{6}$ .

$\frac{4}{6} + 3 - 2$

Step 7.3.9

Cancel the common factor of $4$ and $6$ .

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Step 7.3.9.1

Factor $2$ out of $4$ .

$\frac{2 (2)}{6} + 3 - 2$

Step 7.3.9.2

Cancel the common factors.

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Step 7.3.9.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 2}{2 \cdot 3} + 3 - 2$

Step 7.3.9.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 3} + 3 - 2$

Step 7.3.9.2.3

Rewrite the expression.

$\frac{2}{3} + 3 - 2$

Step 7.3.10

Subtract $2$ from $3$ .

$\frac{2}{3} + 1$

Step 7.3.11

Write $1$ as a fraction with a common denominator.

$\frac{2}{3} + \frac{3}{3}$

Step 7.3.12

Combine the numerators over the common denominator.

$\frac{2 + 3}{3}$

Step 7.3.13

Add $2$ and $3$ .

$\frac{5}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{3}$

Decimal Form:

$1.\overline{6}$

Mixed Number Form:

$1 \frac{2}{3}$

Step 9