Calculus Examples

$\frac{{(x - 1)}^{3}}{x^{3}}$

Step 1

Write $\frac{{(x - 1)}^{3}}{x^{3}}$ as a function.

$f (x) = \frac{{(x - 1)}^{3}}{x^{3}}$

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.

$F (x) = \int \frac{{(x - 1)}^{3}}{x^{3}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int {(x - 1)}^{3} {(x^{3})}^{- 1} d x$

Step 4.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.2.2

Multiply $3$ by $- 1$ .

$\int {(x - 1)}^{3} x^{- 3} d x$

Step 5

Let $u_{1} = x - 1$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 5.1

Let $u_{1} = x - 1$ . Find $\frac{d u_{1}}{d x}$ .

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Step 5.1.1

Differentiate $x - 1$ .

$\frac{d}{d x} [x - 1]$

Step 5.1.2

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 1]$

Step 5.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

Step 5.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int {u_{1}}^{3} {(u_{1} + 1)}^{- 3} d u_{1}$

Step 6

Let $u_{2} = u_{1} + 1$ . Then $d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.1

Let $u_{2} = u_{1} + 1$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 6.1.1

Differentiate $u_{1} + 1$ .

$\frac{d}{d u_{1}} [u_{1} + 1]$

Step 6.1.2

By the Sum Rule, the derivative of $u_{1} + 1$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [1]$ .

$\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [1]$

Step 6.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d u_{1}} [1]$

Step 6.1.4

Since $1$ is constant with respect to $u_{1}$ , the derivative of $1$ with respect to $u_{1}$ is $0$ .

$1 + 0$

Step 6.1.5

Add $1$ and $0$ .

$1$

Step 6.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int {(u_{2} - 1)}^{3} {u_{2}}^{- 3} d u_{2}$

Step 7

Simplify.

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Step 7.1

Use the Binomial Theorem.

$\int ({u_{2}}^{3} + 3 {u_{2}}^{2} \cdot - 1 + 3 u_{2} {(- 1)}^{2} + {(- 1)}^{3}) {u_{2}}^{- 3} d u_{2}$

Step 7.2

Rewrite the exponentiation as a product.

$\int ({u_{2}}^{3} + 3 {u_{2}}^{2} \cdot - 1 + 3 u_{2} (- - 1) + {(- 1)}^{3}) {u_{2}}^{- 3} d u_{2}$

Step 7.3

Rewrite the exponentiation as a product.

$\int ({u_{2}}^{3} + 3 {u_{2}}^{2} \cdot - 1 + 3 u_{2} (- - 1) - {(- 1)}^{2}) {u_{2}}^{- 3} d u_{2}$

Step 7.4

Rewrite the exponentiation as a product.

$\int ({u_{2}}^{3} + 3 {u_{2}}^{2} \cdot - 1 + 3 u_{2} (- - 1) - - - 1) {u_{2}}^{- 3} d u_{2}$

Step 7.5

Apply the distributive property.

$\int ({u_{2}}^{3} + 3 {u_{2}}^{2} \cdot - 1 + 3 u_{2} (- - 1)) {u_{2}}^{- 3} - - - 1 {u_{2}}^{- 3} d u_{2}$

Step 7.6

Apply the distributive property.

$\int ({u_{2}}^{3} + 3 {u_{2}}^{2} \cdot - 1) {u_{2}}^{- 3} + 3 u_{2} (- - 1) {u_{2}}^{- 3} - - - 1 {u_{2}}^{- 3} d u_{2}$

Step 7.7

Apply the distributive property.

$\int {u_{2}}^{3} {u_{2}}^{- 3} + 3 {u_{2}}^{2} \cdot - 1 {u_{2}}^{- 3} + 3 u_{2} (- - 1) {u_{2}}^{- 3} - - - 1 {u_{2}}^{- 3} d u_{2}$

Step 7.8

Move ${u_{2}}^{2}$ .

$\int {u_{2}}^{3} {u_{2}}^{- 3} + 3 \cdot - 1 {u_{2}}^{2} {u_{2}}^{- 3} + 3 u_{2} (- - 1) {u_{2}}^{- 3} - - - 1 {u_{2}}^{- 3} d u_{2}$

Step 7.9

Move $u_{2}$ .

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

Step 7.10

Move parentheses.

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

Step 7.11

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

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

Step 7.12

Subtract $3$ from $3$ .

$\int {u_{2}}^{0} + 3 \cdot - 1 {u_{2}}^{2} {u_{2}}^{- 3} + 3 \cdot - 1 \cdot - 1 u_{2} \cdot {u_{2}}^{- 3} - - 1 \cdot - 1 {u_{2}}^{- 3} d u_{2}$

Step 7.13

Anything raised to $0$ is $1$ .

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

Step 7.14

Multiply $3$ by $- 1$ .

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

Step 7.15

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

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

Step 7.16

Subtract $3$ from $2$ .

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

Step 7.17

Multiply $3$ by $- 1$ .

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

Step 7.18

Multiply $- 3$ by $- 1$ .

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

Step 7.19

Raise $u_{2}$ to the power of $1$ .

$\int 1 - 3 {u_{2}}^{- 1} + 3 ({u_{2}}^{1} {u_{2}}^{- 3}) - - 1 \cdot - 1 {u_{2}}^{- 3} d u_{2}$

Step 7.20

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

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

Step 7.21

Subtract $3$ from $1$ .

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

Step 7.22

Multiply $- 1$ by $- 1$ .

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

Step 7.23

Multiply $- 1$ by $1$ .

$\int 1 - 3 {u_{2}}^{- 1} + 3 {u_{2}}^{- 2} - 1 {u_{2}}^{- 3} d u_{2}$

Step 7.24

Reorder $1$ and $- 3 {u_{2}}^{- 1}$ .

$\int - 3 {u_{2}}^{- 1} + 1 + 3 {u_{2}}^{- 2} - 1 {u_{2}}^{- 3} d u_{2}$

Step 7.25

Move $1$ .

$\int - 3 {u_{2}}^{- 1} + 3 {u_{2}}^{- 2} + 1 - 1 {u_{2}}^{- 3} d u_{2}$

Step 7.26

Move $1$ .

$\int - 3 {u_{2}}^{- 1} + 3 {u_{2}}^{- 2} - 1 {u_{2}}^{- 3} + 1 d u_{2}$

Step 8

Split the single integral into multiple integrals.

$\int - 3 {u_{2}}^{- 1} d u_{2} + \int 3 {u_{2}}^{- 2} d u_{2} + \int - 1 {u_{2}}^{- 3} d u_{2} + \int d u_{2}$

Step 9

Since $- 3$ is constant with respect to $u_{2}$ , move $- 3$ out of the integral.

$- 3 \int {u_{2}}^{- 1} d u_{2} + \int 3 {u_{2}}^{- 2} d u_{2} + \int - 1 {u_{2}}^{- 3} d u_{2} + \int d u_{2}$

Step 10

The integral of ${u_{2}}^{- 1}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$- 3 (\ln (| u_{2} |) + C) + \int 3 {u_{2}}^{- 2} d u_{2} + \int - 1 {u_{2}}^{- 3} d u_{2} + \int d u_{2}$

Step 11

Since $3$ is constant with respect to $u_{2}$ , move $3$ out of the integral.

$- 3 (\ln (| u_{2} |) + C) + 3 \int {u_{2}}^{- 2} d u_{2} + \int - 1 {u_{2}}^{- 3} d u_{2} + \int d u_{2}$

Step 12

By the Power Rule, the integral of ${u_{2}}^{- 2}$ with respect to $u_{2}$ is $- {u_{2}}^{- 1}$ .

$- 3 (\ln (| u_{2} |) + C) + 3 (- {u_{2}}^{- 1} + C) + \int - 1 {u_{2}}^{- 3} d u_{2} + \int d u_{2}$

Step 13

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$- 3 (\ln (| u_{2} |) + C) + 3 (- {u_{2}}^{- 1} + C) - \int {u_{2}}^{- 3} d u_{2} + \int d u_{2}$

Step 14

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

$- 3 (\ln (| u_{2} |) + C) + 3 (- {u_{2}}^{- 1} + C) - (- \frac{1}{2} {u_{2}}^{- 2} + C) + \int d u_{2}$

Step 15

Simplify.

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Step 15.1

Combine ${u_{2}}^{- 2}$ and $\frac{1}{2}$ .

$- 3 (\ln (| u_{2} |) + C) + 3 (- {u_{2}}^{- 1} + C) - (- \frac{{u_{2}}^{- 2}}{2} + C) + \int d u_{2}$

Step 15.2

Move ${u_{2}}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 3 (\ln (| u_{2} |) + C) + 3 (- {u_{2}}^{- 1} + C) - (- \frac{1}{2 {u_{2}}^{2}} + C) + \int d u_{2}$

Step 16

Apply the constant rule.

$- 3 (\ln (| u_{2} |) + C) + 3 (- {u_{2}}^{- 1} + C) - (- \frac{1}{2 {u_{2}}^{2}} + C) + u_{2} + C$

Step 17

Simplify.

$- 3 \ln (| u_{2} |) - \frac{3}{u_{2}} + \frac{1}{2 {u_{2}}^{2}} + u_{2} + C$

Step 18

Substitute back in for each integration substitution variable.

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Step 18.1

Replace all occurrences of $u_{2}$ with $u_{1} + 1$ .

$- 3 \ln (| u_{1} + 1 |) - \frac{3}{u_{1} + 1} + \frac{1}{2 {(u_{1} + 1)}^{2}} + u_{1} + 1 + C$

Step 18.2

Replace all occurrences of $u_{1}$ with $x - 1$ .

$- 3 \ln (| x - 1 + 1 |) - \frac{3}{x - 1 + 1} + \frac{1}{2 {(x - 1 + 1)}^{2}} + x - 1 + 1 + C$

Step 19

Simplify.

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Step 19.1

Add $- 1$ and $1$ .

$- 3 \ln (| x + 0 |) - \frac{3}{x - 1 + 1} + \frac{1}{2 {(x - 1 + 1)}^{2}} + x - 1 + 1 + C$

Step 19.2

Add $x$ and $0$ .

$- 3 \ln (| x |) - \frac{3}{x - 1 + 1} + \frac{1}{2 {(x - 1 + 1)}^{2}} + x - 1 + 1 + C$

Step 19.3

Add $- 1$ and $1$ .

$- 3 \ln (| x |) - \frac{3}{x + 0} + \frac{1}{2 {(x - 1 + 1)}^{2}} + x - 1 + 1 + C$

Step 19.4

Add $x$ and $0$ .

$- 3 \ln (| x |) - \frac{3}{x} + \frac{1}{2 {(x - 1 + 1)}^{2}} + x - 1 + 1 + C$

Step 19.5

Add $- 1$ and $1$ .

$- 3 \ln (| x |) - \frac{3}{x} + \frac{1}{2 {(x + 0)}^{2}} + x - 1 + 1 + C$

Step 19.6

Add $x$ and $0$ .

$- 3 \ln (| x |) - \frac{3}{x} + \frac{1}{2 x^{2}} + x - 1 + 1 + C$

Step 19.7

Add $- 1$ and $1$ .

$- 3 \ln (| x |) - \frac{3}{x} + \frac{1}{2 x^{2}} + x + 0 + C$

Step 19.8

Add $x$ and $0$ .

$- 3 \ln (| x |) - \frac{3}{x} + \frac{1}{2 x^{2}} + x + C$

Step 20

The answer is the antiderivative of the function $f (x) = \frac{{(x - 1)}^{3}}{x^{3}}$ .

$F (x) =$ $- 3 \ln (| x |) - \frac{3}{x} + \frac{1}{2 x^{2}} + x + C$