Calculus Examples

$\sum_{1}^{3} \frac{1}{2} j (2 j^{2} + 3 j + 5)$

Step 1

Simplify the summation.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.2

Rewrite using the commutative property of multiplication.

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

Step 1.2.3

Move $5$ to the left of $j$ .

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

Step 1.3

Simplify each term.

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Step 1.3.1

Multiply $j$ by $j^{2}$ by adding the exponents.

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Step 1.3.1.1

Move $j^{2}$ .

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

Step 1.3.1.2

Multiply $j^{2}$ by $j$ .

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Step 1.3.1.2.1

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

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

Step 1.3.1.2.2

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

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

Step 1.3.1.3

Add $2$ and $1$ .

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

Step 1.3.2

Multiply $j$ by $j$ by adding the exponents.

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Step 1.3.2.1

Move $j$ .

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

Step 1.3.2.2

Multiply $j$ by $j$ .

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

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

Step 1.4

Apply the distributive property.

$\frac{1}{2} (2 j^{3}) + \frac{1}{2} (3 j^{2}) + \frac{1}{2} (5 j)$

Step 1.5

Simplify.

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Step 1.5.1

Cancel the common factor of $2$ .

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Step 1.5.1.1

Factor $2$ out of $2 j^{3}$ .

$\frac{1}{2} (2 (j^{3})) + \frac{1}{2} (3 j^{2}) + \frac{1}{2} (5 j)$

Step 1.5.1.2

Cancel the common factor.

$\frac{1}{2} (2 j^{3}) + \frac{1}{2} (3 j^{2}) + \frac{1}{2} (5 j)$

Step 1.5.1.3

Rewrite the expression.

$j^{3} + \frac{1}{2} (3 j^{2}) + \frac{1}{2} (5 j)$

Step 1.5.2

Multiply $\frac{1}{2} (3 j^{2})$ .

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Step 1.5.2.1

Combine $3$ and $\frac{1}{2}$ .

$j^{3} + \frac{3}{2} j^{2} + \frac{1}{2} (5 j)$

Step 1.5.2.2

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

$j^{3} + \frac{3 j^{2}}{2} + \frac{1}{2} (5 j)$

Step 1.5.3

Multiply $\frac{1}{2} (5 j)$ .

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Step 1.5.3.1

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

$j^{3} + \frac{3 j^{2}}{2} + \frac{5}{2} j$

Step 1.5.3.2

Combine $\frac{5}{2}$ and $j$ .

$j^{3} + \frac{3 j^{2}}{2} + \frac{5 j}{2}$

Step 1.6

Rewrite the summation.

$\sum_{j = 1}^{3} j^{3} + \frac{3 j^{2}}{2} + \frac{5 j}{2}$

Step 2

The formula for the summation of a polynomial with degree $3$ is:

$\sum_{k = 1}^{n} k^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}$

Step 3

Substitute the values into the formula and make sure to multiply by the front term.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Add $3$ and $1$ .

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

Step 4.1.2

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

$(\frac{1}{2} + 5) \frac{9 \cdot 4^{2}}{4}$

Step 4.1.3

Raise $4$ to the power of $2$ .

$(\frac{1}{2} + 5) \frac{9 \cdot 16}{4}$

Step 4.2

Multiply $9$ by $16$ .

$(\frac{1}{2} + 5) \frac{144}{4}$

Step 4.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(\frac{1}{2} + 5 \cdot \frac{2}{2}) \frac{144}{4}$

Step 4.4

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

$(\frac{1}{2} + \frac{5 \cdot 2}{2}) \frac{144}{4}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{1 + 5 \cdot 2}{2} \cdot \frac{144}{4}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $5$ by $2$ .

$\frac{1 + 10}{2} \cdot \frac{144}{4}$

Step 4.6.2

Add $1$ and $10$ .

$\frac{11}{2} \cdot \frac{144}{4}$

Step 4.7

Cancel the common factor of $2$ .

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Step 4.7.1

Factor $2$ out of $144$ .

$\frac{11}{2} \cdot \frac{2 (72)}{4}$

Step 4.7.2

Cancel the common factor.

$\frac{11}{2} \cdot \frac{2 \cdot 72}{4}$

Step 4.7.3

Rewrite the expression.

$11 (\frac{72}{4})$

Step 4.8

Combine $11$ and $\frac{72}{4}$ .

$\frac{11 \cdot 72}{4}$

Step 4.9

Simplify the expression.

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Step 4.9.1

Multiply $11$ by $72$ .

$\frac{792}{4}$

Step 4.9.2

Divide $792$ by $4$ .

$198$