Calculus Examples

$\int (\frac{6 x^{2} - 2 x - 7}{\sqrt{x}}) d x$

Step 1

Remove parentheses.

$\int \frac{6 x^{2} - 2 x - 7}{\sqrt{x}} d x$

Step 2

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

$\int \frac{6 x^{2} - 2 x - 7}{x^{\frac{1}{2}}} d x$

Step 3

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

$\int (6 x^{2} - 2 x - 7) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 4

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

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Step 4.1

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

$\int (6 x^{2} - 2 x - 7) x^{\frac{1}{2} \cdot - 1} d x$

Step 4.2

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

$\int (6 x^{2} - 2 x - 7) x^{\frac{- 1}{2}} d x$

Step 4.3

Move the negative in front of the fraction.

$\int (6 x^{2} - 2 x - 7) x^{- \frac{1}{2}} d x$

Step 5

Expand $(6 x^{2} - 2 x - 7) x^{- \frac{1}{2}}$ .

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Step 5.1

Apply the distributive property.

$\int (6 x^{2} - 2 x) x^{- \frac{1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 5.2

Apply the distributive property.

$\int 6 x^{2} x^{- \frac{1}{2}} - 2 x \cdot x^{- \frac{1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 5.3

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

$\int 6 x^{2 - \frac{1}{2}} - 2 x \cdot x^{- \frac{1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 5.4

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

$\int 6 x^{2 \cdot \frac{2}{2} - \frac{1}{2}} - 2 x \cdot x^{- \frac{1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 5.5

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

$\int 6 x^{\frac{2 \cdot 2}{2} - \frac{1}{2}} - 2 x \cdot x^{- \frac{1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 5.6

Combine the numerators over the common denominator.

$\int 6 x^{\frac{2 \cdot 2 - 1}{2}} - 2 x \cdot x^{- \frac{1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 5.7

Simplify the numerator.

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Step 5.7.1

Multiply $2$ by $2$ .

$\int 6 x^{\frac{4 - 1}{2}} - 2 x \cdot x^{- \frac{1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 5.7.2

Subtract $1$ from $4$ .

$\int 6 x^{\frac{3}{2}} - 2 x \cdot x^{- \frac{1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 5.8

Raise $x$ to the power of $1$ .

$\int 6 x^{\frac{3}{2}} - 2 (x^{1} x^{- \frac{1}{2}}) - 7 x^{- \frac{1}{2}} d x$

Step 5.9

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

$\int 6 x^{\frac{3}{2}} - 2 x^{1 - \frac{1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 5.10

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

$\int 6 x^{\frac{3}{2}} - 2 x^{\frac{2}{2} - \frac{1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 5.11

Combine the numerators over the common denominator.

$\int 6 x^{\frac{3}{2}} - 2 x^{\frac{2 - 1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 5.12

Subtract $1$ from $2$ .

$\int 6 x^{\frac{3}{2}} - 2 x^{\frac{1}{2}} - 7 x^{- \frac{1}{2}} d x$

Step 6

Split the single integral into multiple integrals.

$\int 6 x^{\frac{3}{2}} d x + \int - 2 x^{\frac{1}{2}} d x + \int - 7 x^{- \frac{1}{2}} d x$

Step 7

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int x^{\frac{3}{2}} d x + \int - 2 x^{\frac{1}{2}} d x + \int - 7 x^{- \frac{1}{2}} d x$

Step 8

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

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

Step 9

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

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

Step 10

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

$6 (\frac{2}{5} x^{\frac{5}{2}} + C) - 2 (\frac{2}{3} x^{\frac{3}{2}} + C) + \int - 7 x^{- \frac{1}{2}} d x$

Step 11

Since $- 7$ is constant with respect to $x$ , move $- 7$ out of the integral.

$6 (\frac{2}{5} x^{\frac{5}{2}} + C) - 2 (\frac{2}{3} x^{\frac{3}{2}} + C) - 7 \int x^{- \frac{1}{2}} d x$

Step 12

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

$6 (\frac{2}{5} x^{\frac{5}{2}} + C) - 2 (\frac{2}{3} x^{\frac{3}{2}} + C) - 7 (2 x^{\frac{1}{2}} + C)$

Step 13

Simplify.

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Step 13.1

Simplify.

$\frac{12 x^{\frac{5}{2}}}{5} - \frac{4 x^{\frac{3}{2}}}{3} - 7 (2 x^{\frac{1}{2}}) + C$

Step 13.2

Multiply $2$ by $- 7$ .

$\frac{12 x^{\frac{5}{2}}}{5} - \frac{4 x^{\frac{3}{2}}}{3} - 14 x^{\frac{1}{2}} + C$

Step 14

Reorder terms.

$\frac{12}{5} x^{\frac{5}{2}} - \frac{4}{3} x^{\frac{3}{2}} - 14 x^{\frac{1}{2}} + C$