Calculus Examples

$\int_{0}^{1} y {(y^{2} + 1)}^{5} d y$

Step 1

Let $u = y^{2} + 1$ . Then $d u = 2 y d y$ , so $\frac{1}{2} d u = y d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = y^{2} + 1$ . Find $\frac{d u}{d y}$ .

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Step 1.1.1

Differentiate $y^{2} + 1$ .

$\frac{d}{d y} [y^{2} + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $y^{2} + 1$ with respect to $y$ is $\frac{d}{d y} [y^{2}] + \frac{d}{d y} [1]$ .

$\frac{d}{d y} [y^{2}] + \frac{d}{d y} [1]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$2 y + \frac{d}{d y} [1]$

Step 1.1.4

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$2 y + 0$

Step 1.1.5

Add $2 y$ and $0$ .

$2 y$

Step 1.2

Substitute the lower limit in for $y$ in $u = y^{2} + 1$ .

$u_{lower} = 0^{2} + 1$

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 0 + 1$

Step 1.3.2

Add $0$ and $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $y$ in $u = y^{2} + 1$ .

$u_{upper} = 1^{2} + 1$

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$u_{upper} = 1 + 1$

Step 1.5.2

Add $1$ and $1$ .

$u_{upper} = 2$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 2$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{2} u^{5} \frac{1}{2} d u$

Step 2

Combine $u^{5}$ and $\frac{1}{2}$ .

$\int_{1}^{2} \frac{u^{5}}{2} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{1}^{2} u^{5} d u$

Step 4

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$\frac{1}{2} \frac{1}{6} u^{6}]_{1}^{2}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{6} u^{6}$ at $2$ and at $1$ .

$\frac{1}{2} ((\frac{1}{6} \cdot 2^{6}) - \frac{1}{6} \cdot 1^{6})$

Step 5.2

Simplify.

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Step 5.2.1

Raise $2$ to the power of $6$ .

$\frac{1}{2} (\frac{1}{6} \cdot 64 - \frac{1}{6} \cdot 1^{6})$

Step 5.2.2

Combine $\frac{1}{6}$ and $64$ .

$\frac{1}{2} (\frac{64}{6} - \frac{1}{6} \cdot 1^{6})$

Step 5.2.3

Cancel the common factor of $64$ and $6$ .

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Step 5.2.3.1

Factor $2$ out of $64$ .

$\frac{1}{2} (\frac{2 (32)}{6} - \frac{1}{6} \cdot 1^{6})$

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $2$ out of $6$ .

$\frac{1}{2} (\frac{2 \cdot 32}{2 \cdot 3} - \frac{1}{6} \cdot 1^{6})$

Step 5.2.3.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{2 \cdot 32}{2 \cdot 3} - \frac{1}{6} \cdot 1^{6})$

Step 5.2.3.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{32}{3} - \frac{1}{6} \cdot 1^{6})$

Step 5.2.4

One to any power is one.

$\frac{1}{2} (\frac{32}{3} - \frac{1}{6} \cdot 1)$

Step 5.2.5

Multiply $- 1$ by $1$ .

$\frac{1}{2} (\frac{32}{3} - \frac{1}{6})$

Step 5.2.6

To write $\frac{32}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} (\frac{32}{3} \cdot \frac{2}{2} - \frac{1}{6})$

Step 5.2.7

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.7.1

Multiply $\frac{32}{3}$ by $\frac{2}{2}$ .

$\frac{1}{2} (\frac{32 \cdot 2}{3 \cdot 2} - \frac{1}{6})$

Step 5.2.7.2

Multiply $3$ by $2$ .

$\frac{1}{2} (\frac{32 \cdot 2}{6} - \frac{1}{6})$

Step 5.2.8

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{32 \cdot 2 - 1}{6}$

Step 5.2.9

Simplify the numerator.

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Step 5.2.9.1

Multiply $32$ by $2$ .

$\frac{1}{2} \cdot \frac{64 - 1}{6}$

Step 5.2.9.2

Subtract $1$ from $64$ .

$\frac{1}{2} \cdot \frac{63}{6}$

Step 5.2.10

Cancel the common factor of $63$ and $6$ .

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Step 5.2.10.1

Factor $3$ out of $63$ .

$\frac{1}{2} \cdot \frac{3 (21)}{6}$

Step 5.2.10.2

Cancel the common factors.

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Step 5.2.10.2.1

Factor $3$ out of $6$ .

$\frac{1}{2} \cdot \frac{3 \cdot 21}{3 \cdot 2}$

Step 5.2.10.2.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{3 \cdot 21}{3 \cdot 2}$

Step 5.2.10.2.3

Rewrite the expression.

$\frac{1}{2} \cdot \frac{21}{2}$

Step 5.2.11

Multiply $\frac{1}{2}$ by $\frac{21}{2}$ .

$\frac{21}{2 \cdot 2}$

Step 5.2.12

Multiply $2$ by $2$ .

$\frac{21}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{21}{4}$

Decimal Form:

$5.25$

Mixed Number Form:

$5 \frac{1}{4}$

Step 7