Calculus Examples

$\int_{0}^{2} {(x - x^{2})}^{3} (1 - 2 x) d x$

Step 1

Let $u = x - x^{2}$ . Then $d u = (1 - 2 x) d x$ , so $\frac{1}{1 - 2 x} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x - x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x - x^{2}$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

$1 + \frac{d}{d x} [- x^{2}]$

Step 1.1.3

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

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$1 - \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

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

$1 - (2 x)$

Step 1.1.3.3

Multiply $2$ by $- 1$ .

$1 - 2 x$

Step 1.1.4

Reorder terms.

$- 2 x + 1$

Step 1.1.5

Rearrange terms.

$1 - 2 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = x - x^{2}$ .

$u_{lower} = 0 - 0^{2}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 0 - 0$

Step 1.3.1.2

Multiply $- 1$ by $0$ .

$u_{lower} = 0 + 0$

Step 1.3.2

Add $0$ and $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = x - x^{2}$ .

$u_{upper} = 2 - 2^{2}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Raise $2$ to the power of $2$ .

$u_{upper} = 2 - 1 \cdot 4$

Step 1.5.1.2

Multiply $- 1$ by $4$ .

$u_{upper} = 2 - 4$

Step 1.5.2

Subtract $4$ from $2$ .

$u_{upper} = - 2$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = - 2$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{0}^{- 2} u^{3} d u$

Step 2

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

$\frac{1}{4} u^{4}]_{0}^{- 2}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{1}{4} u^{4}$ at $- 2$ and at $0$ .

$(\frac{1}{4} {(- 2)}^{4}) - \frac{1}{4} \cdot 0^{4}$

Step 3.2

Simplify.

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Step 3.2.1

Raise $- 2$ to the power of $4$ .

$\frac{1}{4} \cdot 16 - \frac{1}{4} \cdot 0^{4}$

Step 3.2.2

Combine $\frac{1}{4}$ and $16$ .

$\frac{16}{4} - \frac{1}{4} \cdot 0^{4}$

Step 3.2.3

Cancel the common factor of $16$ and $4$ .

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Step 3.2.3.1

Factor $4$ out of $16$ .

$\frac{4 \cdot 4}{4} - \frac{1}{4} \cdot 0^{4}$

Step 3.2.3.2

Cancel the common factors.

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Step 3.2.3.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 4}{4 (1)} - \frac{1}{4} \cdot 0^{4}$

Step 3.2.3.2.2

Cancel the common factor.

$\frac{4 \cdot 4}{4 \cdot 1} - \frac{1}{4} \cdot 0^{4}$

Step 3.2.3.2.3

Rewrite the expression.

$\frac{4}{1} - \frac{1}{4} \cdot 0^{4}$

Step 3.2.3.2.4

Divide $4$ by $1$ .

$4 - \frac{1}{4} \cdot 0^{4}$

Step 3.2.4

Raising $0$ to any positive power yields $0$ .

$4 - \frac{1}{4} \cdot 0$

Step 3.2.5

Multiply $0$ by $- 1$ .

$4 + 0 (\frac{1}{4})$

Step 3.2.6

Multiply $0$ by $\frac{1}{4}$ .

$4 + 0$

Step 3.2.7

Add $4$ and $0$ .

$4$

Step 4