Calculus Examples

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

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 2

Expand $x^{2} {(4 x - 5)}^{2}$ .

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Step 2.1

Rewrite ${(4 x - 5)}^{2}$ as $(4 x - 5) (4 x - 5)$ .

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Apply the distributive property.

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

Step 2.6

Apply the distributive property.

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

Step 2.7

Apply the distributive property.

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

Step 2.8

Move $x$ .

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

Step 2.9

Move parentheses.

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

Step 2.10

Move parentheses.

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

Step 2.11

Move $x^{2}$ .

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

Step 2.12

Move $x$ .

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

Step 2.13

Move parentheses.

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

Step 2.14

Move $x^{2}$ .

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

Step 2.15

Move parentheses.

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

Step 2.16

Move $x^{2}$ .

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

Step 2.17

Move $x^{2}$ .

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

Step 2.18

Multiply $4$ by $4$ .

$4 \int 16 x^{2} x \cdot x + 4 \cdot - 5 x^{2} x - 5 \cdot 4 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.19

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

$4 \int 16 (x^{2} x^{1}) x + 4 \cdot - 5 x^{2} x - 5 \cdot 4 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.20

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

$4 \int 16 x^{2 + 1} x + 4 \cdot - 5 x^{2} x - 5 \cdot 4 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.21

Add $2$ and $1$ .

$4 \int 16 x^{3} x + 4 \cdot - 5 x^{2} x - 5 \cdot 4 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.22

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

$4 \int 16 (x^{3} x^{1}) + 4 \cdot - 5 x^{2} x - 5 \cdot 4 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.23

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

$4 \int 16 x^{3 + 1} + 4 \cdot - 5 x^{2} x - 5 \cdot 4 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.24

Add $3$ and $1$ .

$4 \int 16 x^{4} + 4 \cdot - 5 x^{2} x - 5 \cdot 4 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.25

Multiply $4$ by $- 5$ .

$4 \int 16 x^{4} - 20 x^{2} x - 5 \cdot 4 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.26

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

$4 \int 16 x^{4} - 20 (x^{2} x^{1}) - 5 \cdot 4 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.27

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

$4 \int 16 x^{4} - 20 x^{2 + 1} - 5 \cdot 4 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.28

Add $2$ and $1$ .

$4 \int 16 x^{4} - 20 x^{3} - 5 \cdot 4 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.29

Multiply $- 5$ by $4$ .

$4 \int 16 x^{4} - 20 x^{3} - 20 x^{2} x - 5 \cdot - 5 x^{2} d x$

Step 2.30

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

$4 \int 16 x^{4} - 20 x^{3} - 20 (x^{2} x^{1}) - 5 \cdot - 5 x^{2} d x$

Step 2.31

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

$4 \int 16 x^{4} - 20 x^{3} - 20 x^{2 + 1} - 5 \cdot - 5 x^{2} d x$

Step 2.32

Add $2$ and $1$ .

$4 \int 16 x^{4} - 20 x^{3} - 20 x^{3} - 5 \cdot - 5 x^{2} d x$

Step 2.33

Multiply $- 5$ by $- 5$ .

$4 \int 16 x^{4} - 20 x^{3} - 20 x^{3} + 25 x^{2} d x$

Step 2.34

Subtract $20 x^{3}$ from $- 20 x^{3}$ .

$4 \int 16 x^{4} - 40 x^{3} + 25 x^{2} d x$

$4 \int 16 x^{4} - 40 x^{3} + 25 x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$4 (\int 16 x^{4} d x + \int - 40 x^{3} d x + \int 25 x^{2} d x)$

Step 4

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$4 (16 \int x^{4} d x + \int - 40 x^{3} d x + \int 25 x^{2} d x)$

Step 5

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

$4 (16 (\frac{1}{5} x^{5} + C) + \int - 40 x^{3} d x + \int 25 x^{2} d x)$

Step 6

Since $- 40$ is constant with respect to $x$ , move $- 40$ out of the integral.

$4 (16 (\frac{1}{5} x^{5} + C) - 40 \int x^{3} d x + \int 25 x^{2} d x)$

Step 7

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

$4 (16 (\frac{1}{5} x^{5} + C) - 40 (\frac{1}{4} x^{4} + C) + \int 25 x^{2} d x)$

Step 8

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

$4 (16 (\frac{1}{5} x^{5} + C) - 40 (\frac{1}{4} x^{4} + C) + 25 \int x^{2} d x)$

Step 9

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

$4 (16 (\frac{1}{5} x^{5} + C) - 40 (\frac{1}{4} x^{4} + C) + 25 (\frac{1}{3} x^{3} + C))$

Step 10

Simplify.

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Step 10.1

Simplify.

$4 (\frac{16 x^{5}}{5} - 10 x^{4} + \frac{25 x^{3}}{3}) + C$

Step 10.2

Reorder terms.

$4 (\frac{16}{5} x^{5} - 10 x^{4} + \frac{25}{3} x^{3}) + C$

$4 (\frac{16}{5} x^{5} - 10 x^{4} + \frac{25}{3} x^{3}) + C$

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