Calculus Examples

$\int_{1}^{4} 5 y \sqrt{y} + 3 \sqrt{y} d y$

Step 1

Simplify.

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Step 1.1

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

$\int_{1}^{4} 5 y \cdot y^{\frac{1}{2}} + 3 \sqrt{y} d y$

Step 1.2

Multiply $y$ by $y^{\frac{1}{2}}$ by adding the exponents.

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Step 1.2.1

Move $y^{\frac{1}{2}}$ .

$\int_{1}^{4} 5 (y^{\frac{1}{2}} y) + 3 \sqrt{y} d y$

Step 1.2.2

Multiply $y^{\frac{1}{2}}$ by $y$ .

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Step 1.2.2.1

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

$\int_{1}^{4} 5 (y^{\frac{1}{2}} y^{1}) + 3 \sqrt{y} d y$

Step 1.2.2.2

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

$\int_{1}^{4} 5 y^{\frac{1}{2} + 1} + 3 \sqrt{y} d y$

Step 1.2.3

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

$\int_{1}^{4} 5 y^{\frac{1}{2} + \frac{2}{2}} + 3 \sqrt{y} d y$

Step 1.2.4

Combine the numerators over the common denominator.

$\int_{1}^{4} 5 y^{\frac{1 + 2}{2}} + 3 \sqrt{y} d y$

Step 1.2.5

Add $1$ and $2$ .

$\int_{1}^{4} 5 y^{\frac{3}{2}} + 3 \sqrt{y} d y$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{4} 5 y^{\frac{3}{2}} d y + \int_{1}^{4} 3 \sqrt{y} d y$

Step 3

Since $5$ is constant with respect to $y$ , move $5$ out of the integral.

$5 \int_{1}^{4} y^{\frac{3}{2}} d y + \int_{1}^{4} 3 \sqrt{y} d y$

Step 4

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

$5 (\frac{2}{5} y^{\frac{5}{2}}]_{1}^{4}) + \int_{1}^{4} 3 \sqrt{y} d y$

Step 5

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

$5 (\frac{2 y^{\frac{5}{2}}}{5}]_{1}^{4}) + \int_{1}^{4} 3 \sqrt{y} d y$

Step 6

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

$5 (\frac{2 y^{\frac{5}{2}}}{5}]_{1}^{4}) + 3 \int_{1}^{4} \sqrt{y} d y$

Step 7

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

$5 (\frac{2 y^{\frac{5}{2}}}{5}]_{1}^{4}) + 3 \int_{1}^{4} y^{\frac{1}{2}} d y$

Step 8

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

$5 (\frac{2 y^{\frac{5}{2}}}{5}]_{1}^{4}) + 3 (\frac{2}{3} y^{\frac{3}{2}}]_{1}^{4})$

Step 9

Simplify the answer.

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Step 9.1

Combine $\frac{2}{3}$ and $y^{\frac{3}{2}}$ .

$5 (\frac{2 y^{\frac{5}{2}}}{5}]_{1}^{4}) + 3 (\frac{2 y^{\frac{3}{2}}}{3}]_{1}^{4})$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{2 y^{\frac{5}{2}}}{5}$ at $4$ and at $1$ .

$5 ((\frac{2 \cdot 4^{\frac{5}{2}}}{5}) - \frac{2 \cdot 1^{\frac{5}{2}}}{5}) + 3 (\frac{2 y^{\frac{3}{2}}}{3}]_{1}^{4})$

Step 9.2.2

Evaluate $\frac{2 y^{\frac{3}{2}}}{3}$ at $4$ and at $1$ .

$5 (\frac{2 \cdot 4^{\frac{5}{2}}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5}) + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3

Simplify.

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Step 9.2.3.1

Rewrite $4$ as $2^{2}$ .

$5 (\frac{2 \cdot {(2^{2})}^{\frac{5}{2}}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5}) + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.2

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

$5 (\frac{2 \cdot 2^{2 (\frac{5}{2})}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5}) + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.3

Cancel the common factor of $2$ .

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Step 9.2.3.3.1

Cancel the common factor.

$5 (\frac{2 \cdot 2^{2 (\frac{5}{2})}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5}) + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.3.2

Rewrite the expression.

$5 (\frac{2 \cdot 2^{5}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5}) + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.4

Raise $2$ to the power of $5$ .

$5 (\frac{2 \cdot 32}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5}) + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.5

Multiply $2$ by $32$ .

$5 (\frac{64}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5}) + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.6

One to any power is one.

$5 (\frac{64}{5} - \frac{2 \cdot 1}{5}) + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.7

Multiply $2$ by $1$ .

$5 (\frac{64}{5} - \frac{2}{5}) + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.8

Combine the numerators over the common denominator.

$5 \frac{64 - 2}{5} + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.9

Subtract $2$ from $64$ .

$5 (\frac{62}{5}) + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.10

Combine $5$ and $\frac{62}{5}$ .

$\frac{5 \cdot 62}{5} + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.11

Multiply $5$ by $62$ .

$\frac{310}{5} + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.12

Cancel the common factor of $310$ and $5$ .

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Step 9.2.3.12.1

Factor $5$ out of $310$ .

$\frac{5 \cdot 62}{5} + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.12.2

Cancel the common factors.

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Step 9.2.3.12.2.1

Factor $5$ out of $5$ .

$\frac{5 \cdot 62}{5 (1)} + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.12.2.2

Cancel the common factor.

$\frac{5 \cdot 62}{5 \cdot 1} + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.12.2.3

Rewrite the expression.

$\frac{62}{1} + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.12.2.4

Divide $62$ by $1$ .

$62 + 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.13

Rewrite $4$ as $2^{2}$ .

$62 + 3 (\frac{2 \cdot {(2^{2})}^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.14

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

$62 + 3 (\frac{2 \cdot 2^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.15

Cancel the common factor of $2$ .

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Step 9.2.3.15.1

Cancel the common factor.

$62 + 3 (\frac{2 \cdot 2^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.15.2

Rewrite the expression.

$62 + 3 (\frac{2 \cdot 2^{3}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.16

Raise $2$ to the power of $3$ .

$62 + 3 (\frac{2 \cdot 8}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.17

Multiply $2$ by $8$ .

$62 + 3 (\frac{16}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.18

One to any power is one.

$62 + 3 (\frac{16}{3} - \frac{2 \cdot 1}{3})$

Step 9.2.3.19

Multiply $2$ by $1$ .

$62 + 3 (\frac{16}{3} - \frac{2}{3})$

Step 9.2.3.20

Combine the numerators over the common denominator.

$62 + 3 \frac{16 - 2}{3}$

Step 9.2.3.21

Subtract $2$ from $16$ .

$62 + 3 (\frac{14}{3})$

Step 9.2.3.22

Combine $3$ and $\frac{14}{3}$ .

$62 + \frac{3 \cdot 14}{3}$

Step 9.2.3.23

Multiply $3$ by $14$ .

$62 + \frac{42}{3}$

Step 9.2.3.24

Cancel the common factor of $42$ and $3$ .

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Step 9.2.3.24.1

Factor $3$ out of $42$ .

$62 + \frac{3 \cdot 14}{3}$

Step 9.2.3.24.2

Cancel the common factors.

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Step 9.2.3.24.2.1

Factor $3$ out of $3$ .

$62 + \frac{3 \cdot 14}{3 (1)}$

Step 9.2.3.24.2.2

Cancel the common factor.

$62 + \frac{3 \cdot 14}{3 \cdot 1}$

Step 9.2.3.24.2.3

Rewrite the expression.

$62 + \frac{14}{1}$

Step 9.2.3.24.2.4

Divide $14$ by $1$ .

$62 + 14$

Step 9.2.3.25

Add $62$ and $14$ .

$76$

Step 10