Calculus Examples

$\int 2 p y (8 - y^{\frac{3}{2}}) d y$

Step 1

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

$2 p \int y (8 - y^{\frac{3}{2}}) d y$

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

$2 p \int y \cdot 8 + y (- y^{\frac{3}{2}}) d y$

Step 2.2

Reorder $y$ and $8$ .

$2 p \int 8 \cdot y + y (- y^{\frac{3}{2}}) d y$

Step 2.3

Reorder $y$ and $- 1$ .

$2 p \int 8 \cdot y - 1 \cdot y \cdot y^{\frac{3}{2}} d y$

Step 2.4

Factor out negative.

$2 p \int 8 y - (y \cdot y^{\frac{3}{2}}) d y$

Step 2.5

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

$2 p \int 8 y - (y^{1} y^{\frac{3}{2}}) d y$

Step 2.6

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

$2 p \int 8 y - y^{1 + \frac{3}{2}} d y$

Step 2.7

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

$2 p \int 8 y - y^{\frac{2}{2} + \frac{3}{2}} d y$

Step 2.8

Combine the numerators over the common denominator.

$2 p \int 8 y - y^{\frac{2 + 3}{2}} d y$

Step 2.9

Add $2$ and $3$ .

$2 p \int 8 y - y^{\frac{5}{2}} d y$

Step 3

Split the single integral into multiple integrals.

$2 p (\int 8 y d y + \int - y^{\frac{5}{2}} d y)$

Step 4

Since $8$ is constant with respect to $y$ , move $8$ out of the integral.

$2 p (8 \int y d y + \int - y^{\frac{5}{2}} d y)$

Step 5

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

$2 p (8 (\frac{1}{2} y^{2} + C) + \int - y^{\frac{5}{2}} d y)$

Step 6

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

$2 p (8 (\frac{1}{2} y^{2} + C) - \int y^{\frac{5}{2}} d y)$

Step 7

By the Power Rule, the integral of $y^{\frac{5}{2}}$ with respect to $y$ is $\frac{2}{7} y^{\frac{7}{2}}$ .

$2 p (8 (\frac{1}{2} y^{2} + C) - (\frac{2}{7} y^{\frac{7}{2}} + C))$

Step 8

Simplify.

$2 p (4 y^{2} - \frac{2 y^{\frac{7}{2}}}{7}) + C$

Step 9

Reorder terms.

$2 p (4 y^{2} - \frac{2}{7} y^{\frac{7}{2}}) + C$