Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Move the negative in front of the fraction.

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

Step 4

Expand $(3 x^{2} - 2 x + 1) x^{- \frac{1}{2}}$ .

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $2$ by $2$ .

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

Step 4.7.2

Subtract $1$ from $4$ .

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

Combine the numerators over the common denominator.

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

Step 4.12

Subtract $1$ from $2$ .

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

Step 4.13

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

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

Step 4.14

Move $- 2 x^{\frac{1}{2}}$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

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

Step 12.2

Simplify.

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

Step 12.3

Simplify.

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Step 12.3.1

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

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

Step 12.3.2

Multiply $2$ by $- 2$ .

$\frac{6 x^{\frac{5}{2}}}{5} + 2 x^{\frac{1}{2}} + \frac{- 4 x^{\frac{3}{2}}}{3} + C$

Step 12.3.3

Move the negative in front of the fraction.

$\frac{6 x^{\frac{5}{2}}}{5} + 2 x^{\frac{1}{2}} - \frac{4 x^{\frac{3}{2}}}{3} + C$

Step 13

Reorder terms.

$\frac{6}{5} x^{\frac{5}{2}} + 2 x^{\frac{1}{2}} - \frac{4}{3} x^{\frac{3}{2}} + C$