Calculus Examples

$\sum_{x = 1}^{4} {(x - 1)}^{2} + {(x + 1)}^{3}$

Step 1

Simplify the summation.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$(x - 1) (x - 1) + {(x + 1)}^{3}$

Step 1.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$x (x - 1) - 1 (x - 1) + {(x + 1)}^{3}$

Step 1.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 (x - 1) + {(x + 1)}^{3}$

Step 1.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 + {(x + 1)}^{3}$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 1.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 1.1.3.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} - x - x + 1 + {(x + 1)}^{3}$

Step 1.1.3.2

Subtract $x$ from $- x$ .

$x^{2} - 2 x + 1 + {(x + 1)}^{3}$

Step 1.1.4

Use the Binomial Theorem.

$x^{2} - 2 x + 1 + x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3}$

Step 1.1.5

Simplify each term.

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Step 1.1.5.1

Multiply $3$ by $1$ .

$x^{2} - 2 x + 1 + x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3}$

Step 1.1.5.2

One to any power is one.

$x^{2} - 2 x + 1 + x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3}$

Step 1.1.5.3

Multiply $3$ by $1$ .

$x^{2} - 2 x + 1 + x^{3} + 3 x^{2} + 3 x + 1^{3}$

Step 1.1.5.4

One to any power is one.

$x^{2} - 2 x + 1 + x^{3} + 3 x^{2} + 3 x + 1$

Step 1.2

Add $x^{2}$ and $3 x^{2}$ .

$4 x^{2} - 2 x + 1 + x^{3} + 3 x + 1$

Step 1.3

Add $- 2 x$ and $3 x$ .

$4 x^{2} + x + 1 + x^{3} + 1$

Step 1.4

Add $1$ and $1$ .

$4 x^{2} + x + x^{3} + 2$

Step 1.5

Rewrite the summation.

$\sum_{x = 1}^{4} 4 x^{2} + x + x^{3} + 2$

Step 2

Expand the series for each value of $x$ .

$4 \cdot 1^{2} + 1 + 1^{3} + 2 + 4 \cdot 2^{2} + 2 + 2^{3} + 2 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3

Simplify.

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Step 3.1

Raise $1$ to the power of $2$ .

$4 \cdot 1 + 1 + 1^{3} + 2 + 4 \cdot 2^{2} + 2 + 2^{3} + 2 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.2

Multiply $4$ by $1$ .

$4 + 1 + 1^{3} + 2 + 4 \cdot 2^{2} + 2 + 2^{3} + 2 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.3

Add $4$ and $1$ .

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

Step 3.4

Raise $1$ to the power of $3$ .

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

Step 3.5

Add $5$ and $1$ .

$6 + 2 + 4 \cdot 2^{2} + 2 + 2^{3} + 2 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.6

Add $6$ and $2$ .

$8 + 4 \cdot 2^{2} + 2 + 2^{3} + 2 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.7

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

$8 + 4 \cdot 4 + 2 + 2^{3} + 2 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.8

Multiply $4$ by $4$ .

$8 + 16 + 2 + 2^{3} + 2 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.9

Add $16$ and $2$ .

$8 + 18 + 2^{3} + 2 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.10

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

$8 + 18 + 8 + 2 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.11

Add $18$ and $8$ .

$8 + 26 + 2 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.12

Add $26$ and $2$ .

$8 + 28 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.13

Add $8$ and $28$ .

$36 + 4 \cdot 3^{2} + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.14

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

$36 + 4 \cdot 9 + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.15

Multiply $4$ by $9$ .

$36 + 36 + 3 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.16

Add $36$ and $3$ .

$36 + 39 + 3^{3} + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.17

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

$36 + 39 + 27 + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.18

Add $39$ and $27$ .

$36 + 66 + 2 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.19

Add $66$ and $2$ .

$36 + 68 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.20

Add $36$ and $68$ .

$104 + 4 \cdot 4^{2} + 4 + 4^{3} + 2$

Step 3.21

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

$104 + 4 \cdot 16 + 4 + 4^{3} + 2$

Step 3.22

Multiply $4$ by $16$ .

$104 + 64 + 4 + 4^{3} + 2$

Step 3.23

Add $64$ and $4$ .

$104 + 68 + 4^{3} + 2$

Step 3.24

Raise $4$ to the power of $3$ .

$104 + 68 + 64 + 2$

Step 3.25

Add $68$ and $64$ .

$104 + 132 + 2$

Step 3.26

Add $132$ and $2$ .

$104 + 134$

Step 3.27

Add $104$ and $134$ .

$238$