Calculus Examples

$\int (x^{2} + 2) {(x + 1)}^{42} d x$

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 $d u$ .

$\int ({(u - 1)}^{2} + 2) u^{42} d u$

$\int ({(u - 1)}^{2} + 2) u^{42} d u$

Step 2

Expand $({(u - 1)}^{2} + 2) u^{42}$ .

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Step 2.1

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

$\int ((u - 1) (u - 1) + 2) u^{42} d u$

Step 2.2

Apply the distributive property.

$\int (u (u - 1) - 1 (u - 1) + 2) u^{42} d u$

Step 2.3

Apply the distributive property.

$\int (u \cdot u + u \cdot - 1 - 1 (u - 1) + 2) u^{42} d u$

Step 2.4

Apply the distributive property.

$\int (u \cdot u + u \cdot - 1 - 1 u - 1 \cdot - 1 + 2) u^{42} d u$

Step 2.5

Apply the distributive property.

$\int (u \cdot u + u \cdot - 1 - 1 u - 1 \cdot - 1) u^{42} + 2 u^{42} d u$

Step 2.6

Apply the distributive property.

$\int (u \cdot u + u \cdot - 1) u^{42} + (- 1 u - 1 \cdot - 1) u^{42} + 2 u^{42} d u$

Step 2.7

Apply the distributive property.

$\int u \cdot u \cdot u^{42} + u \cdot - 1 u^{42} + (- 1 u - 1 \cdot - 1) u^{42} + 2 u^{42} d u$

Step 2.8

Apply the distributive property.

$\int u \cdot u \cdot u^{42} + u \cdot - 1 u^{42} - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.9

Reorder $u$ and $- 1$ .

$\int u \cdot u \cdot u^{42} - 1 \cdot u \cdot u^{42} - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.10

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

$\int u^{1} u \cdot u^{42} - 1 u \cdot u^{42} - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.11

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

$\int u^{1} u^{1} u^{42} - 1 u \cdot u^{42} - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.12

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

$\int u^{1 + 1} u^{42} - 1 u \cdot u^{42} - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.13

Add $1$ and $1$ .

$\int u^{2} u^{42} - 1 u \cdot u^{42} - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.14

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

$\int u^{2 + 42} - 1 u \cdot u^{42} - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.15

Add $2$ and $42$ .

$\int u^{44} - 1 u \cdot u^{42} - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.16

Factor out negative.

$\int u^{44} - (u \cdot u^{42}) - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.17

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

$\int u^{44} - (u^{1} u^{42}) - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.18

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

$\int u^{44} - u^{1 + 42} - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.19

Add $1$ and $42$ .

$\int u^{44} - u^{43} - 1 u \cdot u^{42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.20

Factor out negative.

$\int u^{44} - u^{43} - (u \cdot u^{42}) - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.21

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

$\int u^{44} - u^{43} - (u^{1} u^{42}) - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.22

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

$\int u^{44} - u^{43} - u^{1 + 42} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.23

Add $1$ and $42$ .

$\int u^{44} - u^{43} - u^{43} - 1 \cdot - 1 u^{42} + 2 u^{42} d u$

Step 2.24

Multiply $- 1$ by $- 1$ .

$\int u^{44} - u^{43} - u^{43} + 1 u^{42} + 2 u^{42} d u$

Step 2.25

Multiply $u^{42}$ by $1$ .

$\int u^{44} - u^{43} - u^{43} + u^{42} + 2 u^{42} d u$

Step 2.26

Subtract $u^{43}$ from $- u^{43}$ .

$\int u^{44} - 2 u^{43} + u^{42} + 2 u^{42} d u$

Step 2.27

Add $u^{42}$ and $2 u^{42}$ .

$\int u^{44} - 2 u^{43} + 3 u^{42} d u$

$\int u^{44} - 2 u^{43} + 3 u^{42} d u$

Step 3

Split the single integral into multiple integrals.

$\int u^{44} d u + \int - 2 u^{43} d u + \int 3 u^{42} d u$

Step 4

By the Power Rule, the integral of $u^{44}$ with respect to $u$ is $\frac{1}{45} u^{45}$ .

$\frac{1}{45} u^{45} + C + \int - 2 u^{43} d u + \int 3 u^{42} d u$

Step 5

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

$\frac{1}{45} u^{45} + C - 2 \int u^{43} d u + \int 3 u^{42} d u$

Step 6

By the Power Rule, the integral of $u^{43}$ with respect to $u$ is $\frac{1}{44} u^{44}$ .

$\frac{1}{45} u^{45} + C - 2 (\frac{1}{44} u^{44} + C) + \int 3 u^{42} d u$

Step 7

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

$\frac{1}{45} u^{45} + C - 2 (\frac{1}{44} u^{44} + C) + 3 \int u^{42} d u$

Step 8

By the Power Rule, the integral of $u^{42}$ with respect to $u$ is $\frac{1}{43} u^{43}$ .

$\frac{1}{45} u^{45} + C - 2 (\frac{1}{44} u^{44} + C) + 3 (\frac{1}{43} u^{43} + C)$

Step 9

Simplify.

$\frac{u^{45}}{45} - \frac{u^{44}}{22} + 3 (\frac{1}{43} u^{43}) + C$

Step 10

Reorder terms.

$\frac{1}{45} u^{45} - \frac{1}{22} u^{44} + 3 (\frac{1}{43}) u^{43} + C$

Step 11

Combine $3$ and $\frac{1}{43}$ .

$\frac{1}{45} u^{45} - \frac{1}{22} u^{44} + \frac{3}{43} u^{43} + C$

Step 12

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

$\frac{1}{45} {(x + 1)}^{45} - \frac{1}{22} {(x + 1)}^{44} + \frac{3}{43} {(x + 1)}^{43} + C$

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