Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

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

Step 1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 y^{3} = 2 \cdot y^{2} \cdot y$

Step 1.1.3

Rewrite the polynomial.

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

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y^{2}$ and $b = y$ .

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

Step 1.2

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

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

Step 1.3

Cancel the common factor of $2$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Simplify.

$\int y (y^{2} - y) d y$

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Reorder $y$ and $- 1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $2$ .

$\int y^{3} - 1 y \cdot y d y$

Step 2.6

Factor out negative.

$\int y^{3} - (y \cdot y) d y$

Step 2.7

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

$\int y^{3} - (y^{1} y) d y$

Step 2.8

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

$\int y^{3} - (y^{1} y^{1}) d y$

Step 2.9

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

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

Step 2.10

Add $1$ and $1$ .

$\int y^{3} - y^{2} d y$

Step 3

Split the single integral into multiple integrals.

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

Step 4

By the Power Rule, the integral of $y^{3}$ with respect to $y$ is $\frac{1}{4} y^{4}$ .

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

Step 5

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

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

Step 6

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

$\frac{1}{4} y^{4} + C - (\frac{1}{3} y^{3} + C)$

Step 7

Simplify.

$\frac{1}{4} y^{4} - \frac{1}{3} y^{3} + C$