Calculus Examples

$\frac{1 - \sqrt{u}}{\sqrt{u}}$

Step 1

Write $\frac{1 - \sqrt{u}}{\sqrt{u}}$ as a function.

$f (u) = \frac{1 - \sqrt{u}}{\sqrt{u}}$

Step 2

The function $F (u)$ can be found by finding the indefinite integral of the derivative $f (u)$ .

$F (u) = \int f (u) d u$

Step 3

Set up the integral to solve.

$F (u) = \int \frac{1 - \sqrt{u}}{\sqrt{u}} d u$

Step 4

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

$\int \frac{1 - u^{\frac{1}{2}}}{\sqrt{u}} d u$

Step 5

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

$\int \frac{1 - u^{\frac{1}{2}}}{u^{\frac{1}{2}}} d u$

Step 6

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

$\int (1 - u^{\frac{1}{2}}) {(u^{\frac{1}{2}})}^{- 1} d u$

Step 7

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

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Step 7.1

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

$\int (1 - u^{\frac{1}{2}}) u^{\frac{1}{2} \cdot - 1} d u$

Step 7.2

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

$\int (1 - u^{\frac{1}{2}}) u^{\frac{- 1}{2}} d u$

Step 7.3

Move the negative in front of the fraction.

$\int (1 - u^{\frac{1}{2}}) u^{- \frac{1}{2}} d u$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\int 1 u^{- \frac{1}{2}} - u^{\frac{1}{2}} u^{- \frac{1}{2}} d u$

Step 8.2

Multiply $u^{- \frac{1}{2}}$ by $1$ .

$\int u^{- \frac{1}{2}} - u^{\frac{1}{2}} u^{- \frac{1}{2}} d u$

Step 8.3

Factor out negative.

$\int u^{- \frac{1}{2}} - (u^{\frac{1}{2}} u^{- \frac{1}{2}}) d u$

Step 8.4

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

$\int u^{- \frac{1}{2}} - u^{\frac{1}{2} - \frac{1}{2}} d u$

Step 8.5

Combine the numerators over the common denominator.

$\int u^{- \frac{1}{2}} - u^{\frac{1 - 1}{2}} d u$

Step 8.6

Subtract $1$ from $1$ .

$\int u^{- \frac{1}{2}} - u^{\frac{0}{2}} d u$

Step 8.7

Cancel the common factor of $0$ and $2$ .

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Step 8.7.1

Factor $2$ out of $0$ .

$\int u^{- \frac{1}{2}} - u^{\frac{2 (0)}{2}} d u$

Step 8.7.2

Cancel the common factors.

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Step 8.7.2.1

Factor $2$ out of $2$ .

$\int u^{- \frac{1}{2}} - u^{\frac{2 \cdot 0}{2 \cdot 1}} d u$

Step 8.7.2.2

Cancel the common factor.

$\int u^{- \frac{1}{2}} - u^{\frac{2 \cdot 0}{2 \cdot 1}} d u$

Step 8.7.2.3

Rewrite the expression.

$\int u^{- \frac{1}{2}} - u^{\frac{0}{1}} d u$

Step 8.7.2.4

Divide $0$ by $1$ .

$\int u^{- \frac{1}{2}} - u^{0} d u$

Step 8.8

Anything raised to $0$ is $1$ .

$\int u^{- \frac{1}{2}} - 1 \cdot 1 d u$

Step 8.9

Multiply $- 1$ by $1$ .

$\int u^{- \frac{1}{2}} - 1 d u$

Step 9

Split the single integral into multiple integrals.

$\int u^{- \frac{1}{2}} d u + \int - 1 d u$

Step 10

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

$2 u^{\frac{1}{2}} + C + \int - 1 d u$

Step 11

Apply the constant rule.

$2 u^{\frac{1}{2}} + C - u + C$

Step 12

Simplify.

$2 u^{\frac{1}{2}} - u + C$

Step 13

The answer is the antiderivative of the function $f (u) = \frac{1 - \sqrt{u}}{\sqrt{u}}$ .

$F (u) =$ $2 u^{\frac{1}{2}} - u + C$