Calculus Examples

$x {(4 x - 1)}^{4}$

Step 1

Write $x {(4 x - 1)}^{4}$ as a function.

$f (x) = x {(4 x - 1)}^{4}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x {(4 x - 1)}^{4} d x$

Step 4

Simplify.

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Step 4.1

Use the Binomial Theorem.

$\int x ({(4 x)}^{4} + 4 {(4 x)}^{3} \cdot - 1 + 6 {(4 x)}^{2} {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2

Simplify each term.

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Step 4.2.1

Apply the product rule to $4 x$ .

$\int x (4^{4} x^{4} + 4 {(4 x)}^{3} \cdot - 1 + 6 {(4 x)}^{2} {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.2

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

$\int x (256 x^{4} + 4 {(4 x)}^{3} \cdot - 1 + 6 {(4 x)}^{2} {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.3

Apply the product rule to $4 x$ .

$\int x (256 x^{4} + 4 (4^{3} x^{3}) \cdot - 1 + 6 {(4 x)}^{2} {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.4

Multiply $4$ by $4^{3}$ by adding the exponents.

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Step 4.2.4.1

Move $4^{3}$ .

$\int x (256 x^{4} + 4^{3} \cdot 4 x^{3} \cdot - 1 + 6 {(4 x)}^{2} {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.4.2

Multiply $4^{3}$ by $4$ .

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Step 4.2.4.2.1

Raise $4$ to the power of $1$ .

$\int x (256 x^{4} + 4^{3} \cdot 4^{1} x^{3} \cdot - 1 + 6 {(4 x)}^{2} {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.4.2.2

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

$\int x (256 x^{4} + 4^{3 + 1} x^{3} \cdot - 1 + 6 {(4 x)}^{2} {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.4.3

Add $3$ and $1$ .

$\int x (256 x^{4} + 4^{4} x^{3} \cdot - 1 + 6 {(4 x)}^{2} {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.5

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

$\int x (256 x^{4} + 256 x^{3} \cdot - 1 + 6 {(4 x)}^{2} {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.6

Multiply $- 1$ by $256$ .

$\int x (256 x^{4} - 256 x^{3} + 6 {(4 x)}^{2} {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.7

Apply the product rule to $4 x$ .

$\int x (256 x^{4} - 256 x^{3} + 6 (4^{2} x^{2}) {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.8

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

$\int x (256 x^{4} - 256 x^{3} + 6 (16 x^{2}) {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.9

Multiply $16$ by $6$ .

$\int x (256 x^{4} - 256 x^{3} + 96 x^{2} {(- 1)}^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.10

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

$\int x (256 x^{4} - 256 x^{3} + 96 x^{2} \cdot 1 + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.11

Multiply $96$ by $1$ .

$\int x (256 x^{4} - 256 x^{3} + 96 x^{2} + 4 (4 x) {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.12

Multiply $4$ by $4$ .

$\int x (256 x^{4} - 256 x^{3} + 96 x^{2} + 16 x {(- 1)}^{3} + {(- 1)}^{4}) d x$

Step 4.2.13

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

$\int x (256 x^{4} - 256 x^{3} + 96 x^{2} + 16 x \cdot - 1 + {(- 1)}^{4}) d x$

Step 4.2.14

Multiply $- 1$ by $16$ .

$\int x (256 x^{4} - 256 x^{3} + 96 x^{2} - 16 x + {(- 1)}^{4}) d x$

Step 4.2.15

Raise $- 1$ to the power of $4$ .

$\int x (256 x^{4} - 256 x^{3} + 96 x^{2} - 16 x + 1) d x$

Step 4.3

Apply the distributive property.

$\int x (256 x^{4}) + x (- 256 x^{3}) + x (96 x^{2}) + x (- 16 x) + x \cdot 1 d x$

Step 4.4

Simplify.

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Step 4.4.1

Rewrite using the commutative property of multiplication.

$\int 256 x \cdot x^{4} + x (- 256 x^{3}) + x (96 x^{2}) + x (- 16 x) + x \cdot 1 d x$

Step 4.4.2

Rewrite using the commutative property of multiplication.

$\int 256 x \cdot x^{4} - 256 x \cdot x^{3} + x (96 x^{2}) + x (- 16 x) + x \cdot 1 d x$

Step 4.4.3

Rewrite using the commutative property of multiplication.

$\int 256 x \cdot x^{4} - 256 x \cdot x^{3} + 96 x \cdot x^{2} + x (- 16 x) + x \cdot 1 d x$

Step 4.4.4

Rewrite using the commutative property of multiplication.

$\int 256 x \cdot x^{4} - 256 x \cdot x^{3} + 96 x \cdot x^{2} - 16 x \cdot x + x \cdot 1 d x$

Step 4.4.5

Multiply $x$ by $1$ .

$\int 256 x \cdot x^{4} - 256 x \cdot x^{3} + 96 x \cdot x^{2} - 16 x \cdot x + x d x$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 4.5.1.1

Move $x^{4}$ .

$\int 256 (x^{4} x) - 256 x \cdot x^{3} + 96 x \cdot x^{2} - 16 x \cdot x + x d x$

Step 4.5.1.2

Multiply $x^{4}$ by $x$ .

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Step 4.5.1.2.1

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

$\int 256 (x^{4} x^{1}) - 256 x \cdot x^{3} + 96 x \cdot x^{2} - 16 x \cdot x + x d x$

Step 4.5.1.2.2

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

$\int 256 x^{4 + 1} - 256 x \cdot x^{3} + 96 x \cdot x^{2} - 16 x \cdot x + x d x$

Step 4.5.1.3

Add $4$ and $1$ .

$\int 256 x^{5} - 256 x \cdot x^{3} + 96 x \cdot x^{2} - 16 x \cdot x + x d x$

Step 4.5.2

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 4.5.2.1

Move $x^{3}$ .

$\int 256 x^{5} - 256 (x^{3} x) + 96 x \cdot x^{2} - 16 x \cdot x + x d x$

Step 4.5.2.2

Multiply $x^{3}$ by $x$ .

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Step 4.5.2.2.1

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

$\int 256 x^{5} - 256 (x^{3} x^{1}) + 96 x \cdot x^{2} - 16 x \cdot x + x d x$

Step 4.5.2.2.2

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

$\int 256 x^{5} - 256 x^{3 + 1} + 96 x \cdot x^{2} - 16 x \cdot x + x d x$

Step 4.5.2.3

Add $3$ and $1$ .

$\int 256 x^{5} - 256 x^{4} + 96 x \cdot x^{2} - 16 x \cdot x + x d x$

Step 4.5.3

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

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Step 4.5.3.1

Move $x^{2}$ .

$\int 256 x^{5} - 256 x^{4} + 96 (x^{2} x) - 16 x \cdot x + x d x$

Step 4.5.3.2

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

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Step 4.5.3.2.1

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

$\int 256 x^{5} - 256 x^{4} + 96 (x^{2} x^{1}) - 16 x \cdot x + x d x$

Step 4.5.3.2.2

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

$\int 256 x^{5} - 256 x^{4} + 96 x^{2 + 1} - 16 x \cdot x + x d x$

Step 4.5.3.3

Add $2$ and $1$ .

$\int 256 x^{5} - 256 x^{4} + 96 x^{3} - 16 x \cdot x + x d x$

Step 4.5.4

Multiply $x$ by $x$ by adding the exponents.

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Step 4.5.4.1

Move $x$ .

$\int 256 x^{5} - 256 x^{4} + 96 x^{3} - 16 (x \cdot x) + x d x$

Step 4.5.4.2

Multiply $x$ by $x$ .

$\int 256 x^{5} - 256 x^{4} + 96 x^{3} - 16 x^{2} + x d x$

Step 5

Split the single integral into multiple integrals.

$\int 256 x^{5} d x + \int - 256 x^{4} d x + \int 96 x^{3} d x + \int - 16 x^{2} d x + \int x d x$

Step 6

Since $256$ is constant with respect to $x$ , move $256$ out of the integral.

$256 \int x^{5} d x + \int - 256 x^{4} d x + \int 96 x^{3} d x + \int - 16 x^{2} d x + \int x d x$

Step 7

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$256 (\frac{1}{6} x^{6} + C) + \int - 256 x^{4} d x + \int 96 x^{3} d x + \int - 16 x^{2} d x + \int x d x$

Step 8

Since $- 256$ is constant with respect to $x$ , move $- 256$ out of the integral.

$256 (\frac{1}{6} x^{6} + C) - 256 \int x^{4} d x + \int 96 x^{3} d x + \int - 16 x^{2} d x + \int x d x$

Step 9

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

$256 (\frac{1}{6} x^{6} + C) - 256 (\frac{1}{5} x^{5} + C) + \int 96 x^{3} d x + \int - 16 x^{2} d x + \int x d x$

Step 10

Since $96$ is constant with respect to $x$ , move $96$ out of the integral.

$256 (\frac{1}{6} x^{6} + C) - 256 (\frac{1}{5} x^{5} + C) + 96 \int x^{3} d x + \int - 16 x^{2} d x + \int x d x$

Step 11

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

$256 (\frac{1}{6} x^{6} + C) - 256 (\frac{1}{5} x^{5} + C) + 96 (\frac{1}{4} x^{4} + C) + \int - 16 x^{2} d x + \int x d x$

Step 12

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

$256 (\frac{1}{6} x^{6} + C) - 256 (\frac{1}{5} x^{5} + C) + 96 (\frac{1}{4} x^{4} + C) - 16 \int x^{2} d x + \int x d x$

Step 13

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

$256 (\frac{1}{6} x^{6} + C) - 256 (\frac{1}{5} x^{5} + C) + 96 (\frac{1}{4} x^{4} + C) - 16 (\frac{1}{3} x^{3} + C) + \int x d x$

Step 14

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

$256 (\frac{1}{6} x^{6} + C) - 256 (\frac{1}{5} x^{5} + C) + 96 (\frac{1}{4} x^{4} + C) - 16 (\frac{1}{3} x^{3} + C) + \frac{1}{2} x^{2} + C$

Step 15

Simplify.

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Step 15.1

Simplify.

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Step 15.1.1

Combine $\frac{1}{6}$ and $x^{6}$ .

$256 (\frac{x^{6}}{6} + C) - 256 (\frac{1}{5} x^{5} + C) + 96 (\frac{1}{4} x^{4} + C) - 16 (\frac{1}{3} x^{3} + C) + \frac{1}{2} x^{2} + C$

Step 15.1.2

Combine $\frac{1}{5}$ and $x^{5}$ .

$256 (\frac{x^{6}}{6} + C) - 256 (\frac{x^{5}}{5} + C) + 96 (\frac{1}{4} x^{4} + C) - 16 (\frac{1}{3} x^{3} + C) + \frac{1}{2} x^{2} + C$

Step 15.1.3

Combine $\frac{1}{4}$ and $x^{4}$ .

$256 (\frac{x^{6}}{6} + C) - 256 (\frac{x^{5}}{5} + C) + 96 (\frac{x^{4}}{4} + C) - 16 (\frac{1}{3} x^{3} + C) + \frac{1}{2} x^{2} + C$

Step 15.1.4

Combine $\frac{1}{3}$ and $x^{3}$ .

$256 (\frac{x^{6}}{6} + C) - 256 (\frac{x^{5}}{5} + C) + 96 (\frac{x^{4}}{4} + C) - 16 (\frac{x^{3}}{3} + C) + \frac{1}{2} x^{2} + C$

Step 15.2

Simplify.

$\frac{128 x^{6}}{3} - \frac{256 x^{5}}{5} + 24 x^{4} - \frac{16 x^{3}}{3} + \frac{1}{2} x^{2} + C$

Step 16

Reorder terms.

$\frac{128}{3} x^{6} - \frac{256}{5} x^{5} + 24 x^{4} - \frac{16}{3} x^{3} + \frac{1}{2} x^{2} + C$

Step 17

The answer is the antiderivative of the function $f (x) = x {(4 x - 1)}^{4}$ .

$F (x) =$ $\frac{128}{3} x^{6} - \frac{256}{5} x^{5} + 24 x^{4} - \frac{16}{3} x^{3} + \frac{1}{2} x^{2} + C$