Calculus Examples

$\int 2 x^{2} {(x - 5)}^{2} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x^{2} {(x - 5)}^{2} d x$

Step 2

Expand $x^{2} {(x - 5)}^{2}$ .

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Step 2.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$2 \int x^{2} ((x - 5) (x - 5)) d x$

Step 2.2

Apply the distributive property.

$2 \int x^{2} (x (x - 5) - 5 (x - 5)) d x$

Step 2.3

Apply the distributive property.

$2 \int x^{2} (x \cdot x + x \cdot - 5 - 5 (x - 5)) d x$

Step 2.4

Apply the distributive property.

$2 \int x^{2} (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) d x$

Step 2.5

Apply the distributive property.

$2 \int x^{2} (x \cdot x + x \cdot - 5) + x^{2} (- 5 x - 5 \cdot - 5) d x$

Step 2.6

Apply the distributive property.

$2 \int x^{2} (x \cdot x) + x^{2} (x \cdot - 5) + x^{2} (- 5 x - 5 \cdot - 5) d x$

Step 2.7

Apply the distributive property.

$2 \int x^{2} (x \cdot x) + x^{2} (x \cdot - 5) + x^{2} (- 5 x) + x^{2} (- 5 \cdot - 5) d x$

Step 2.8

Reorder $x$ and $- 5$ .

$2 \int x^{2} x \cdot x + x^{2} (- 5 \cdot x) + x^{2} (- 5 x) + x^{2} (- 5 \cdot - 5) d x$

Step 2.9

Reorder $x^{2}$ and $- 5$ .

$2 \int x^{2} x \cdot x - 5 \cdot x^{2} \cdot x + x^{2} (- 5 x) + x^{2} (- 5 \cdot - 5) d x$

Step 2.10

Reorder $x^{2}$ and $- 5$ .

$2 \int x^{2} x \cdot x - 5 \cdot x^{2} \cdot x - 5 \cdot x^{2} x + x^{2} (- 5 \cdot - 5) d x$

Step 2.11

Move $x^{2}$ .

$2 \int x^{2} x \cdot x - 5 \cdot x^{2} \cdot x - 5 \cdot x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.12

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

$2 \int x^{2} x^{1} x - 5 x^{2} x - 5 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.13

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

$2 \int x^{2 + 1} x - 5 x^{2} x - 5 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.14

Add $2$ and $1$ .

$2 \int x^{3} x - 5 x^{2} x - 5 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.15

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

$2 \int x^{3} x^{1} - 5 x^{2} x - 5 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.16

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

$2 \int x^{3 + 1} - 5 x^{2} x - 5 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.17

Add $3$ and $1$ .

$2 \int x^{4} - 5 x^{2} x - 5 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.18

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

$2 \int x^{4} - 5 (x^{2} x^{1}) - 5 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.19

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

$2 \int x^{4} - 5 x^{2 + 1} - 5 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.20

Add $2$ and $1$ .

$2 \int x^{4} - 5 x^{3} - 5 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.21

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

$2 \int x^{4} - 5 x^{3} - 5 (x^{2} x^{1}) - 5 \cdot - 5 x^{2} d x$

Step 2.22

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

$2 \int x^{4} - 5 x^{3} - 5 x^{2 + 1} - 5 \cdot - 5 x^{2} d x$

Step 2.23

Add $2$ and $1$ .

$2 \int x^{4} - 5 x^{3} - 5 x^{3} - 5 \cdot - 5 x^{2} d x$

Step 2.24

Multiply $- 5$ by $- 5$ .

$2 \int x^{4} - 5 x^{3} - 5 x^{3} + 25 x^{2} d x$

Step 2.25

Subtract $5 x^{3}$ from $- 5 x^{3}$ .

$2 \int x^{4} - 10 x^{3} + 25 x^{2} d x$

$2 \int x^{4} - 10 x^{3} + 25 x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$2 (\int x^{4} d x + \int - 10 x^{3} d x + \int 25 x^{2} d x)$

Step 4

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$2 (\frac{1}{5} x^{5} + C + \int - 10 x^{3} d x + \int 25 x^{2} d x)$

Step 5

Since $- 10$ is constant with respect to $x$ , move $- 10$ out of the integral.

$2 (\frac{1}{5} x^{5} + C - 10 \int x^{3} d x + \int 25 x^{2} d x)$

Step 6

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$2 (\frac{1}{5} x^{5} + C - 10 (\frac{1}{4} x^{4} + C) + \int 25 x^{2} d x)$

Step 7

Since $25$ is constant with respect to $x$ , move $25$ out of the integral.

$2 (\frac{1}{5} x^{5} + C - 10 (\frac{1}{4} x^{4} + C) + 25 \int x^{2} d x)$

Step 8

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

$2 (\frac{1}{5} x^{5} + C - 10 (\frac{1}{4} x^{4} + C) + 25 (\frac{1}{3} x^{3} + C))$

Step 9

Simplify.

$2 (\frac{x^{5}}{5} - \frac{5 x^{4}}{2} + \frac{25 x^{3}}{3}) + C$

Step 10

Reorder terms.

$2 (\frac{1}{5} x^{5} - \frac{5}{2} x^{4} + \frac{25}{3} x^{3}) + C$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$