Calculus Examples

$\int_{0}^{1} y^{3} - yd y d y$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{1} y^{3} d y + \int_{0}^{1} - yd y d y$

Step 2

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

$\frac{1}{4} y^{4}]_{0}^{1} + \int_{0}^{1} - yd y d y$

Step 3

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

$\frac{1}{4} y^{4}]_{0}^{1} - yd \int_{0}^{1} y d y$

Step 4

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

$\frac{1}{4} y^{4}]_{0}^{1} - yd \frac{1}{2} y^{2}]_{0}^{1}$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{2}$ and $y^{2}$ .

$\frac{1}{4} y^{4}]_{0}^{1} - yd \frac{y^{2}}{2}]_{0}^{1}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{1}{4} y^{4}$ at $1$ and at $0$ .

$(\frac{1}{4} \cdot 1^{4}) - \frac{1}{4} \cdot 0^{4} - yd \frac{y^{2}}{2}]_{0}^{1}$

Step 5.2.2

Evaluate $\frac{y^{2}}{2}$ at $1$ and at $0$ .

$\frac{1}{4} \cdot 1^{4} - \frac{1}{4} \cdot 0^{4} - yd (\frac{1^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3

Simplify.

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Step 5.2.3.1

One to any power is one.

$\frac{1}{4} \cdot 1 - \frac{1}{4} \cdot 0^{4} - yd (\frac{1^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.2

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4} - \frac{1}{4} \cdot 0^{4} - yd (\frac{1^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.3

Raising $0$ to any positive power yields $0$ .

$\frac{1}{4} - \frac{1}{4} \cdot 0 - yd (\frac{1^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.4

Multiply $0$ by $- 1$ .

$\frac{1}{4} + 0 (\frac{1}{4}) - yd (\frac{1^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.5

Multiply $0$ by $\frac{1}{4}$ .

$\frac{1}{4} + 0 - yd (\frac{1^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.6

Add $\frac{1}{4}$ and $0$ .

$\frac{1}{4} - yd (\frac{1^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.7

One to any power is one.

$\frac{1}{4} - yd (\frac{1}{2} - \frac{0^{2}}{2})$

Step 5.2.3.8

Raising $0$ to any positive power yields $0$ .

$\frac{1}{4} - yd (\frac{1}{2} - \frac{0}{2})$

Step 5.2.3.9

Cancel the common factor of $0$ and $2$ .

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Step 5.2.3.9.1

Factor $2$ out of $0$ .

$\frac{1}{4} - yd (\frac{1}{2} - \frac{2 (0)}{2})$

Step 5.2.3.9.2

Cancel the common factors.

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Step 5.2.3.9.2.1

Factor $2$ out of $2$ .

$\frac{1}{4} - yd (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.2.3.9.2.2

Cancel the common factor.

$\frac{1}{4} - yd (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.2.3.9.2.3

Rewrite the expression.

$\frac{1}{4} - yd (\frac{1}{2} - \frac{0}{1})$

Step 5.2.3.9.2.4

Divide $0$ by $1$ .

$\frac{1}{4} - yd (\frac{1}{2} - 0)$

Step 5.2.3.10

Multiply $- 1$ by $0$ .

$\frac{1}{4} - yd (\frac{1}{2} + 0)$

Step 5.2.3.11

Add $\frac{1}{2}$ and $0$ .

$\frac{1}{4} - yd \frac{1}{2}$

Step 5.3

Simplify.

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Step 5.3.1

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

$\frac{1}{4} + \frac{- 1}{2} yd$

Step 5.3.2

Move the negative in front of the fraction.

$\frac{1}{4} - \frac{1}{2} yd$

Step 6