Calculus Examples

$\frac{7 \sqrt{x} - 3 x^{2} - 3}{4 \sqrt{x}}$

Step 1

Write $\frac{7 \sqrt{x} - 3 x^{2} - 3}{4 \sqrt{x}}$ as a function.

$f (x) = \frac{7 \sqrt{x} - 3 x^{2} - 3}{4 \sqrt{x}}$

Step 2

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

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

Step 3

Set up the integral to solve.

$F (x) = \int \frac{7 \sqrt{x} - 3 x^{2} - 3}{4 \sqrt{x}} d x$

Step 4

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int \frac{7 \sqrt{x} - 3 x^{2} - 3}{\sqrt{x}} d x$

Step 5

Apply basic rules of exponents.

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Step 5.1

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

$\frac{1}{4} \int \frac{7 x^{\frac{1}{2}} - 3 x^{2} - 3}{\sqrt{x}} d x$

Step 5.2

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

$\frac{1}{4} \int \frac{7 x^{\frac{1}{2}} - 3 x^{2} - 3}{x^{\frac{1}{2}}} d x$

Step 5.3

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

$\frac{1}{4} \int (7 x^{\frac{1}{2}} - 3 x^{2} - 3) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 5.4

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

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Step 5.4.1

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

$\frac{1}{4} \int (7 x^{\frac{1}{2}} - 3 x^{2} - 3) x^{\frac{1}{2} \cdot - 1} d x$

Step 5.4.2

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

$\frac{1}{4} \int (7 x^{\frac{1}{2}} - 3 x^{2} - 3) x^{\frac{- 1}{2}} d x$

Step 5.4.3

Move the negative in front of the fraction.

$\frac{1}{4} \int (7 x^{\frac{1}{2}} - 3 x^{2} - 3) x^{- \frac{1}{2}} d x$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{1}{4} \int (7 x^{\frac{1}{2}} - 3 x^{2}) x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.2

Apply the distributive property.

$\frac{1}{4} \int 7 x^{\frac{1}{2}} x^{- \frac{1}{2}} - 3 x^{2} x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.3

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

$\frac{1}{4} \int 7 x^{\frac{1}{2} - \frac{1}{2}} - 3 x^{2} x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.4

Combine the numerators over the common denominator.

$\frac{1}{4} \int 7 x^{\frac{1 - 1}{2}} - 3 x^{2} x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.5

Subtract $1$ from $1$ .

$\frac{1}{4} \int 7 x^{\frac{0}{2}} - 3 x^{2} x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.6

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

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Step 6.6.1

Factor $2$ out of $0$ .

$\frac{1}{4} \int 7 x^{\frac{2 (0)}{2}} - 3 x^{2} x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.6.2

Cancel the common factors.

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Step 6.6.2.1

Factor $2$ out of $2$ .

$\frac{1}{4} \int 7 x^{\frac{2 \cdot 0}{2 \cdot 1}} - 3 x^{2} x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.6.2.2

Cancel the common factor.

$\frac{1}{4} \int 7 x^{\frac{2 \cdot 0}{2 \cdot 1}} - 3 x^{2} x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.6.2.3

Rewrite the expression.

$\frac{1}{4} \int 7 x^{\frac{0}{1}} - 3 x^{2} x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.6.2.4

Divide $0$ by $1$ .

$\frac{1}{4} \int 7 x^{0} - 3 x^{2} x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.7

Anything raised to $0$ is $1$ .

$\frac{1}{4} \int 7 \cdot 1 - 3 x^{2} x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.8

Multiply $7$ by $1$ .

$\frac{1}{4} \int 7 - 3 x^{2} x^{- \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.9

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

$\frac{1}{4} \int 7 - 3 x^{2 - \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.10

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

$\frac{1}{4} \int 7 - 3 x^{2 \cdot \frac{2}{2} - \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.11

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

$\frac{1}{4} \int 7 - 3 x^{\frac{2 \cdot 2}{2} - \frac{1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.12

Combine the numerators over the common denominator.

$\frac{1}{4} \int 7 - 3 x^{\frac{2 \cdot 2 - 1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.13

Simplify the numerator.

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Step 6.13.1

Multiply $2$ by $2$ .

$\frac{1}{4} \int 7 - 3 x^{\frac{4 - 1}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.13.2

Subtract $1$ from $4$ .

$\frac{1}{4} \int 7 - 3 x^{\frac{3}{2}} - 3 x^{- \frac{1}{2}} d x$

Step 6.14

Reorder $7$ and $- 3 x^{\frac{3}{2}}$ .

$\frac{1}{4} \int - 3 x^{\frac{3}{2}} + 7 - 3 x^{- \frac{1}{2}} d x$

Step 6.15

Move $7$ .

$\frac{1}{4} \int - 3 x^{\frac{3}{2}} - 3 x^{- \frac{1}{2}} + 7 d x$

Step 7

Split the single integral into multiple integrals.

$\frac{1}{4} (\int - 3 x^{\frac{3}{2}} d x + \int - 3 x^{- \frac{1}{2}} d x + \int 7 d x)$

Step 8

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$\frac{1}{4} (- 3 \int x^{\frac{3}{2}} d x + \int - 3 x^{- \frac{1}{2}} d x + \int 7 d x)$

Step 9

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$\frac{1}{4} (- 3 (\frac{2}{5} x^{\frac{5}{2}} + C) + \int - 3 x^{- \frac{1}{2}} d x + \int 7 d x)$

Step 10

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$\frac{1}{4} (- 3 (\frac{2}{5} x^{\frac{5}{2}} + C) - 3 \int x^{- \frac{1}{2}} d x + \int 7 d x)$

Step 11

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

$\frac{1}{4} (- 3 (\frac{2}{5} x^{\frac{5}{2}} + C) - 3 (2 x^{\frac{1}{2}} + C) + \int 7 d x)$

Step 12

Apply the constant rule.

$\frac{1}{4} (- 3 (\frac{2}{5} x^{\frac{5}{2}} + C) - 3 (2 x^{\frac{1}{2}} + C) + 7 x + C)$

Step 13

Simplify.

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Step 13.1

Combine $\frac{2}{5}$ and $x^{\frac{5}{2}}$ .

$\frac{1}{4} (- 3 (\frac{2 x^{\frac{5}{2}}}{5} + C) - 3 (2 x^{\frac{1}{2}} + C) + 7 x + C)$

Step 13.2

Simplify.

$\frac{1}{4} (- \frac{6 x^{\frac{5}{2}}}{5} - 6 x^{\frac{1}{2}} + 7 x) + C$

Step 14

Reorder terms.

$\frac{1}{4} (- \frac{6}{5} x^{\frac{5}{2}} - 6 x^{\frac{1}{2}} + 7 x) + C$

Step 15

The answer is the antiderivative of the function $f (x) = \frac{7 \sqrt{x} - 3 x^{2} - 3}{4 \sqrt{x}}$ .

$F (x) =$ $\frac{1}{4} (- \frac{6}{5} x^{\frac{5}{2}} - 6 x^{\frac{1}{2}} + 7 x) + C$