Calculus Examples

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

Step 1

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x + 1$ .

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

Step 1.1.2

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 1.1.3

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 1.1.4

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

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

$1$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $3$ by $- 1$ .

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

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

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

Step 3

Expand $({(u - 1)}^{2} - 3) u^{- 3}$ .

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Step 3.1

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

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Apply the distributive property.

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

Step 3.6

Apply the distributive property.

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

Step 3.7

Apply the distributive property.

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

Step 3.8

Apply the distributive property.

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

Step 3.9

Reorder $u$ and $- 1$ .

$\int u \cdot u \cdot u^{- 3} - 1 \cdot u \cdot u^{- 3} - 1 u \cdot u^{- 3} - 1 \cdot - 1 u^{- 3} - 3 u^{- 3} d u$

Step 3.10

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

$\int u^{1} u \cdot u^{- 3} - 1 u \cdot u^{- 3} - 1 u \cdot u^{- 3} - 1 \cdot - 1 u^{- 3} - 3 u^{- 3} d u$

Step 3.11

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

$\int u^{1} u^{1} u^{- 3} - 1 u \cdot u^{- 3} - 1 u \cdot u^{- 3} - 1 \cdot - 1 u^{- 3} - 3 u^{- 3} d u$

Step 3.12

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

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

Step 3.13

Add $1$ and $1$ .

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

Step 3.14

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

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

Step 3.15

Subtract $3$ from $2$ .

$\int u^{- 1} - 1 u \cdot u^{- 3} - 1 u \cdot u^{- 3} - 1 \cdot - 1 u^{- 3} - 3 u^{- 3} d u$

Step 3.16

Factor out negative.

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

Step 3.17

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

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

Step 3.18

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

$\int u^{- 1} - u^{1 - 3} - 1 u \cdot u^{- 3} - 1 \cdot - 1 u^{- 3} - 3 u^{- 3} d u$

Step 3.19

Subtract $3$ from $1$ .

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

Step 3.20

Factor out negative.

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

Step 3.21

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

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

Step 3.22

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

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

Step 3.23

Subtract $3$ from $1$ .

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

Step 3.24

Multiply $- 1$ by $- 1$ .

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

Step 3.25

Multiply $u^{- 3}$ by $1$ .

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

Step 3.26

Subtract $u^{- 2}$ from $- u^{- 2}$ .

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

Step 3.27

Subtract $3 u^{- 3}$ from $u^{- 3}$ .

$\int u^{- 1} - 2 u^{- 2} - 2 u^{- 3} d u$

$\int u^{- 1} - 2 u^{- 2} - 2 u^{- 3} d u$

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

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

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

Step 10.1.2

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

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

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

Step 10.2

Simplify.

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

Step 10.3

Rewrite $\ln (| u |) + \frac{2}{u} - 2 (- \frac{1}{2 u^{2}}) + C$ as $\ln (| u |) + \frac{2}{u} - \frac{- 1}{u^{2}} + C$ .

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

Step 10.4

Simplify.

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Step 10.4.1

Move the negative in front of the fraction.

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

Step 10.4.2

Multiply $- 1$ by $- 1$ .

$\ln (| u |) + \frac{2}{u} + 1 \frac{1}{u^{2}} + C$

Step 10.4.3

Multiply $\frac{1}{u^{2}}$ by $1$ .

$\ln (| u |) + \frac{2}{u} + \frac{1}{u^{2}} + C$

$\ln (| u |) + \frac{2}{u} + \frac{1}{u^{2}} + C$

$\ln (| u |) + \frac{2}{u} + \frac{1}{u^{2}} + C$

Step 11

Replace all occurrences of $u$ with $x + 1$ .

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

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