Calculus Examples

$3 {(6 - 5 x - 2 x^{2} + 7 x^{4})}^{2} (- 5 - 4 x + 28 x^{3})$

Step 1

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

$3 \int {(6 - 5 x - 2 x^{2} + 7 x^{4})}^{2} (- 5 - 4 x + 28 x^{3}) d x$

Step 2

Let $u = 6 - 5 x - 2 x^{2} + 7 x^{4}$ . Then $d u = (- 5 - 4 x + 28 x^{3}) d x$ , so $\frac{1}{- 5 - 4 x + 28 x^{3}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 6 - 5 x - 2 x^{2} + 7 x^{4}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $6 - 5 x - 2 x^{2} + 7 x^{4}$ .

$\frac{d}{d x} [6 - 5 x - 2 x^{2} + 7 x^{4}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $6 - 5 x - 2 x^{2} + 7 x^{4}$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [7 x^{4}]$ .

$\frac{d}{d x} [6] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [7 x^{4}]$

Step 2.1.2.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [7 x^{4}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 2.1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$0 - 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [7 x^{4}]$

Step 2.1.3.2

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

$0 - 5 \cdot 1 + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [7 x^{4}]$

Step 2.1.3.3

Multiply $- 5$ by $1$ .

$0 - 5 + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [7 x^{4}]$

Step 2.1.4

Evaluate $\frac{d}{d x} [- 2 x^{2}]$ .

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Step 2.1.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{2}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{2}]$ .

$0 - 5 - 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [7 x^{4}]$

Step 2.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - 5 - 2 (2 x) + \frac{d}{d x} [7 x^{4}]$

Step 2.1.4.3

Multiply $2$ by $- 2$ .

$0 - 5 - 4 x + \frac{d}{d x} [7 x^{4}]$

Step 2.1.5

Evaluate $\frac{d}{d x} [7 x^{4}]$ .

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Step 2.1.5.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{4}$ with respect to $x$ is $7 \frac{d}{d x} [x^{4}]$ .

$0 - 5 - 4 x + 7 \frac{d}{d x} [x^{4}]$

Step 2.1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$0 - 5 - 4 x + 7 (4 x^{3})$

Step 2.1.5.3

Multiply $4$ by $7$ .

$0 - 5 - 4 x + 28 x^{3}$

Step 2.1.6

Simplify.

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Step 2.1.6.1

Subtract $5$ from $0$ .

$- 5 - 4 x + 28 x^{3}$

Step 2.1.6.2

Reorder terms.

$28 x^{3} - 4 x - 5$

Step 2.1.7

Rearrange terms.

$- 4 x - 5 + 28 x^{3}$

Step 2.1.8

Rearrange terms.

$- 5 - 4 x + 28 x^{3}$

Step 2.2

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

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

Step 3

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

$3 (\frac{1}{3} u^{3} + C)$

Step 4

Simplify.

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Step 4.1

Rewrite $3 (\frac{1}{3} u^{3} + C)$ as $3 (\frac{1}{3}) u^{3} + C$ .

$3 (\frac{1}{3}) u^{3} + C$

Step 4.2

Simplify.

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Step 4.2.1

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

$\frac{3}{3} u^{3} + C$

Step 4.2.2

Cancel the common factor of $3$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{3}{3} u^{3} + C$

Step 4.2.2.2

Rewrite the expression.

$1 u^{3} + C$

Step 4.2.3

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

$u^{3} + C$

Step 5

Replace all occurrences of $u$ with $6 - 5 x - 2 x^{2} + 7 x^{4}$ .

${(6 - 5 x - 2 x^{2} + 7 x^{4})}^{3} + C$