Calculus Examples

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

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 $d u$ .

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Apply the distributive property.

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

Step 2.6

Apply the distributive property.

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

Step 2.7

Apply the distributive property.

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

Step 2.8

Reorder $u$ and $- 1$ .

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Add $1$ and $1$ .

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

Step 2.13

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

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

Step 2.14

Add $2$ and $3$ .

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

Step 2.15

Factor out negative.

$\int u^{5} - (u \cdot u^{3}) - 1 u \cdot u^{3} - 1 \cdot - 1 u^{3} d u$

Step 2.16

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

$\int u^{5} - (u^{1} u^{3}) - 1 u \cdot u^{3} - 1 \cdot - 1 u^{3} d u$

Step 2.17

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

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

Step 2.18

Add $1$ and $3$ .

$\int u^{5} - u^{4} - 1 u \cdot u^{3} - 1 \cdot - 1 u^{3} d u$

Step 2.19

Factor out negative.

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

Step 2.20

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

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

Step 2.21

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

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

Step 2.22

Add $1$ and $3$ .

$\int u^{5} - u^{4} - u^{4} - 1 \cdot - 1 u^{3} d u$

Step 2.23

Multiply $- 1$ by $- 1$ .

$\int u^{5} - u^{4} - u^{4} + 1 u^{3} d u$

Step 2.24

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

$\int u^{5} - u^{4} - u^{4} + u^{3} d u$

Step 2.25

Subtract $u^{4}$ from $- u^{4}$ .

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

$\frac{1}{6} u^{6} + C + \int - 2 u^{4} d u + \int u^{3} d u$

Step 5

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

$\frac{1}{6} u^{6} + C - 2 \int u^{4} d u + \int u^{3} d u$

Step 6

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

$\frac{1}{6} u^{6} + C - 2 (\frac{1}{5} u^{5} + C) + \int u^{3} d u$

Step 7

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

$\frac{1}{6} u^{6} + C - 2 (\frac{1}{5} u^{5} + C) + \frac{1}{4} u^{4} + C$

Step 8

Simplify.

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Step 8.1

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

$\frac{1}{6} u^{6} + C - 2 (\frac{u^{5}}{5} + C) + \frac{1}{4} u^{4} + C$

Step 8.2

Simplify.

$\frac{u^{6}}{6} - \frac{2 u^{5}}{5} + \frac{1}{4} u^{4} + C$

$\frac{u^{6}}{6} - \frac{2 u^{5}}{5} + \frac{1}{4} u^{4} + C$

Step 9

Reorder terms.

$\frac{1}{6} u^{6} - \frac{2}{5} u^{5} + \frac{1}{4} u^{4} + C$

Step 10

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

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

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