Calculus Examples

$\int_{1}^{\sqrt{11}} \frac{s^{2} + \sqrt{s}}{s^{2}} d s$

Step 1

Simplify.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{s}$ as $s^{\frac{1}{2}}$ .

$\int_{1}^{\sqrt{11}} \frac{s^{2} + s^{\frac{1}{2}}}{s^{2}} d s$

Step 1.1.2

Factor $s^{\frac{1}{2}}$ out of $s^{2} + s^{\frac{1}{2}}$ .

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Step 1.1.2.1

Factor $s^{\frac{1}{2}}$ out of $s^{2}$ .

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} s^{\frac{3}{2}} + s^{\frac{1}{2}}}{s^{2}} d s$

Step 1.1.2.2

Multiply by $1$ .

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} s^{\frac{3}{2}} + s^{\frac{1}{2}} \cdot 1}{s^{2}} d s$

Step 1.1.2.3

Factor $s^{\frac{1}{2}}$ out of $s^{\frac{1}{2}} s^{\frac{3}{2}} + s^{\frac{1}{2}} \cdot 1$ .

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} (s^{\frac{3}{2}} + 1)}{s^{2}} d s$

Step 1.1.3

Rewrite $s^{\frac{3}{2}}$ as ${(s^{\frac{1}{2}})}^{3}$ .

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} ({(s^{\frac{1}{2}})}^{3} + 1)}{s^{2}} d s$

Step 1.1.4

Rewrite $1$ as $1^{3}$ .

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} ({(s^{\frac{1}{2}})}^{3} + 1^{3})}{s^{2}} d s$

Step 1.1.5

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = s^{\frac{1}{2}}$ and $b = 1$ .

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} ((s^{\frac{1}{2}} + 1) ({(s^{\frac{1}{2}})}^{2} - s^{\frac{1}{2}} \cdot 1 + 1^{2}))}{s^{2}} d s$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Multiply the exponents in ${(s^{\frac{1}{2}})}^{2}$ .

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Step 1.1.6.1.1

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

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} ((s^{\frac{1}{2}} + 1) (s^{\frac{1}{2} \cdot 2} - s^{\frac{1}{2}} \cdot 1 + 1^{2}))}{s^{2}} d s$

Step 1.1.6.1.2

Cancel the common factor of $2$ .

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Step 1.1.6.1.2.1

Cancel the common factor.

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} ((s^{\frac{1}{2}} + 1) (s^{\frac{1}{2} \cdot 2} - s^{\frac{1}{2}} \cdot 1 + 1^{2}))}{s^{2}} d s$

Step 1.1.6.1.2.2

Rewrite the expression.

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} ((s^{\frac{1}{2}} + 1) (s^{1} - s^{\frac{1}{2}} \cdot 1 + 1^{2}))}{s^{2}} d s$

Step 1.1.6.2

Simplify.

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} ((s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} \cdot 1 + 1^{2}))}{s^{2}} d s$

Step 1.1.6.3

Multiply $- 1$ by $1$ .

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} ((s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1^{2}))}{s^{2}} d s$

Step 1.1.6.4

One to any power is one.

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} ((s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1))}{s^{2}} d s$

$\int_{1}^{\sqrt{11}} \frac{s^{\frac{1}{2}} (s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1)}{s^{2}} d s$

Step 1.2

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

$\int_{1}^{\sqrt{11}} \frac{(s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1)}{s^{2} s^{- \frac{1}{2}}} d s$

Step 1.3

Multiply $s^{2}$ by $s^{- \frac{1}{2}}$ by adding the exponents.

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Step 1.3.1

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

$\int_{1}^{\sqrt{11}} \frac{(s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1)}{s^{2 - \frac{1}{2}}} d s$

Step 1.3.2

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

$\int_{1}^{\sqrt{11}} \frac{(s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1)}{s^{2 \cdot \frac{2}{2} - \frac{1}{2}}} d s$

Step 1.3.3

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

$\int_{1}^{\sqrt{11}} \frac{(s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1)}{s^{\frac{2 \cdot 2}{2} - \frac{1}{2}}} d s$

Step 1.3.4

Combine the numerators over the common denominator.

$\int_{1}^{\sqrt{11}} \frac{(s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1)}{s^{\frac{2 \cdot 2 - 1}{2}}} d s$

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Multiply $2$ by $2$ .

$\int_{1}^{\sqrt{11}} \frac{(s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1)}{s^{\frac{4 - 1}{2}}} d s$

Step 1.3.5.2

Subtract $1$ from $4$ .

$\int_{1}^{\sqrt{11}} \frac{(s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1)}{s^{\frac{3}{2}}} d s$

Step 2

Move $s^{\frac{3}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int_{1}^{\sqrt{11}} (s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1) {(s^{\frac{3}{2}})}^{- 1} d s$

Step 3

Multiply the exponents in ${(s^{\frac{3}{2}})}^{- 1}$ .

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Step 3.1

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

$\int_{1}^{\sqrt{11}} (s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1) s^{\frac{3}{2} \cdot - 1} d s$

Step 3.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 3.2.1

Combine $\frac{3}{2}$ and $- 1$ .

$\int_{1}^{\sqrt{11}} (s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1) s^{\frac{3 \cdot - 1}{2}} d s$

Step 3.2.2

Multiply $3$ by $- 1$ .

$\int_{1}^{\sqrt{11}} (s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1) s^{\frac{- 3}{2}} d s$

Step 3.3

Move the negative in front of the fraction.

$\int_{1}^{\sqrt{11}} (s^{\frac{1}{2}} + 1) (s - s^{\frac{1}{2}} + 1) s^{- \frac{3}{2}} d s$

Step 4

Simplify.

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Step 4.1