Calculus Examples

$\sum_{i = 1}^{10} i (i^{2} + 1)$

Step 1

Simplify the summation.

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Step 1.1

Apply the distributive property.

$i \cdot i^{2} + i \cdot 1$

Step 1.2

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

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Step 1.2.1

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

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Step 1.2.1.1

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

$i^{1} i^{2} + i \cdot 1$

Step 1.2.1.2

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

$i^{1 + 2} + i \cdot 1$

Step 1.2.2

Add $1$ and $2$ .

$i^{3} + i \cdot 1$

Step 1.3

Multiply $i$ by $1$ .

$i^{3} + i$

Step 1.4

Rewrite the summation.

$\sum_{i = 1}^{10} i^{3} + i$

Step 2

Split the summation into smaller summations that fit the summation rules.

$\sum_{i = 1}^{10} i^{3} + i = \sum_{i = 1}^{10} i^{3} + \sum_{i = 1}^{10} i$

Step 3

Evaluate $\sum_{i = 1}^{10} i^{3}$ .

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Step 3.1

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

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

Step 3.2

Substitute the values into the formula.

$\frac{10^{2} {(10 + 1)}^{2}}{4}$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Add $10$ and $1$ .

$\frac{10^{2} \cdot 11^{2}}{4}$

Step 3.3.1.2

Raise $10$ to the power of $2$ .

$\frac{100 \cdot 11^{2}}{4}$

Step 3.3.1.3

Raise $11$ to the power of $2$ .

$\frac{100 \cdot 121}{4}$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Multiply $100$ by $121$ .

$\frac{12100}{4}$

Step 3.3.2.2

Divide $12100$ by $4$ .

$3025$

Step 4

Evaluate $\sum_{i = 1}^{10} i$ .

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Step 4.1

The formula for the summation of a polynomial with degree $1$ is:

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

Step 4.2

Substitute the values into the formula.

$\frac{10 (10 + 1)}{2}$

Step 4.3

Simplify.

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Step 4.3.1

Cancel the common factor of $10$ and $2$ .

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Step 4.3.1.1

Factor $2$ out of $10 (10 + 1)$ .

$\frac{2 (5 (10 + 1))}{2}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $2$ .

$\frac{2 (5 (10 + 1))}{2 (1)}$

Step 4.3.1.2.2

Cancel the common factor.

$\frac{2 (5 (10 + 1))}{2 \cdot 1}$

Step 4.3.1.2.3

Rewrite the expression.

$\frac{5 (10 + 1)}{1}$

Step 4.3.1.2.4

Divide $5 (10 + 1)$ by $1$ .

$5 (10 + 1)$

Step 4.3.2

Simplify the expression.

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Step 4.3.2.1

Add $10$ and $1$ .

$5 \cdot 11$

Step 4.3.2.2

Multiply $5$ by $11$ .

$55$

Step 5

Add the results of the summations.

$3025 + 55$

Step 6

Add $3025$ and $55$ .

$3080$