Calculus Examples

$\int \frac{y^{4} + 2 y^{2} - 1}{\sqrt{y}} d y$

Step 1

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

$\int \frac{y^{4} + 2 y^{2} - 1}{y^{\frac{1}{2}}} d y$

Step 2

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

$\int (y^{4} + 2 y^{2} - 1) {(y^{\frac{1}{2}})}^{- 1} d y$

Step 3

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

Tap for more steps...

Step 3.1

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

$\int (y^{4} + 2 y^{2} - 1) y^{\frac{1}{2} \cdot - 1} d y$

Step 3.2

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

$\int (y^{4} + 2 y^{2} - 1) y^{\frac{- 1}{2}} d y$

Step 3.3

Move the negative in front of the fraction.

$\int (y^{4} + 2 y^{2} - 1) y^{- \frac{1}{2}} d y$

Step 4

Expand $(y^{4} + 2 y^{2} - 1) y^{- \frac{1}{2}}$ .

Tap for more steps...

Step 4.1

Apply the distributive property.

$\int (y^{4} + 2 y^{2}) y^{- \frac{1}{2}} - 1 y^{- \frac{1}{2}} d y$

Step 4.2

Apply the distributive property.

$\int y^{4} y^{- \frac{1}{2}} + 2 y^{2} y^{- \frac{1}{2}} - 1 y^{- \frac{1}{2}} d y$

Step 4.3

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

$\int y^{4 - \frac{1}{2}} + 2 y^{2} y^{- \frac{1}{2}} - 1 y^{- \frac{1}{2}} d y$

Step 4.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.5

Combine $4$ and $\frac{2}{2}$ .

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 4.7

Simplify the numerator.

Tap for more steps...

Step 4.7.1

Multiply $4$ by $2$ .

$\int y^{\frac{8 - 1}{2}} + 2 y^{2} y^{- \frac{1}{2}} - 1 y^{- \frac{1}{2}} d y$

Step 4.7.2

Subtract $1$ from $8$ .

$\int y^{\frac{7}{2}} + 2 y^{2} y^{- \frac{1}{2}} - 1 y^{- \frac{1}{2}} d y$

Step 4.8

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

$\int y^{\frac{7}{2}} + 2 y^{2 - \frac{1}{2}} - 1 y^{- \frac{1}{2}} d y$

Step 4.9

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

$\int y^{\frac{7}{2}} + 2 y^{2 \cdot \frac{2}{2} - \frac{1}{2}} - 1 y^{- \frac{1}{2}} d y$

Step 4.10

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

$\int y^{\frac{7}{2}} + 2 y^{\frac{2 \cdot 2}{2} - \frac{1}{2}} - 1 y^{- \frac{1}{2}} d y$

Step 4.11

Combine the numerators over the common denominator.

$\int y^{\frac{7}{2}} + 2 y^{\frac{2 \cdot 2 - 1}{2}} - 1 y^{- \frac{1}{2}} d y$

Step 4.12

Simplify the numerator.

Tap for more steps...

Step 4.12.1

Multiply $2$ by $2$ .

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

Step 4.12.2

Subtract $1$ from $4$ .

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

Step 4.13

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

By the Power Rule, the integral of $y^{\frac{7}{2}}$ with respect to $y$ is $\frac{2}{9} y^{\frac{9}{2}}$ .

$2 (\frac{2}{5} y^{\frac{5}{2}} + C) + \frac{2}{9} y^{\frac{9}{2}} + C + \int - 1 y^{- \frac{1}{2}} d y$

Step 9

Simplify.

Tap for more steps...

Step 9.1

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

$2 (\frac{2 y^{\frac{5}{2}}}{5} + C) + \frac{2}{9} y^{\frac{9}{2}} + C + \int - 1 y^{- \frac{1}{2}} d y$

Step 9.2

Rewrite $- 1 y^{- \frac{1}{2}}$ as $- y^{- \frac{1}{2}}$ .

$2 (\frac{2 y^{\frac{5}{2}}}{5} + C) + \frac{2}{9} y^{\frac{9}{2}} + C + \int - y^{- \frac{1}{2}} d y$

Step 10

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

$2 (\frac{2 y^{\frac{5}{2}}}{5} + C) + \frac{2}{9} y^{\frac{9}{2}} + C - \int y^{- \frac{1}{2}} d y$

Step 11

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

$2 (\frac{2 y^{\frac{5}{2}}}{5} + C) + \frac{2}{9} y^{\frac{9}{2}} + C - (2 y^{\frac{1}{2}} + C)$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Simplify.

$\frac{4 y^{\frac{5}{2}}}{5} + \frac{2 y^{\frac{9}{2}}}{9} - (2 y^{\frac{1}{2}}) + C$

Step 12.2

Simplify the expression.

Tap for more steps...

Step 12.2.1

Multiply $2$ by $- 1$ .

$\frac{4 y^{\frac{5}{2}}}{5} + \frac{2 y^{\frac{9}{2}}}{9} - 2 y^{\frac{1}{2}} + C$

Step 12.2.2

Reorder terms.

$\frac{4}{5} y^{\frac{5}{2}} + \frac{2}{9} y^{\frac{9}{2}} - 2 y^{\frac{1}{2}} + C$