Calculus Examples

$\int_{0}^{1} 2 x^{2} u (x) d x = 1$

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

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

$\int_{0}^{1} 2 u x^{2 + 1} d x$

Step 1.3

Add $2$ and $1$ .

$\int_{0}^{1} 2 u x^{3} d x$

Step 2

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

$2 u \int_{0}^{1} x^{3} d x$

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{1}{4}$ and $x^{4}$ .

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

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{x^{4}}{4}$ at $1$ and at $0$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

One to any power is one.

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Cancel the common factor of $0$ and $4$ .

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Step 4.2.2.3.1

Factor $4$ out of $0$ .

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

Step 4.2.2.3.2

Cancel the common factors.

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Step 4.2.2.3.2.1

Factor $4$ out of $4$ .

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

Step 4.2.2.3.2.2

Cancel the common factor.

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

Step 4.2.2.3.2.3

Rewrite the expression.

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

Step 4.2.2.3.2.4

Divide $0$ by $1$ .

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

Step 4.2.2.4

Multiply $- 1$ by $0$ .

$2 u (\frac{1}{4} + 0)$

Step 4.2.2.5

Add $\frac{1}{4}$ and $0$ .

$2 u \frac{1}{4}$

Step 4.2.2.6

Combine $2$ and $\frac{1}{4}$ .

$u \frac{2}{4}$

Step 4.2.2.7

Combine $u$ and $\frac{2}{4}$ .

$\frac{u \cdot 2}{4}$

Step 4.2.2.8

Move $2$ to the left of $u$ .

$\frac{2 \cdot u}{4}$

Step 4.2.2.9

Cancel the common factor of $2$ and $4$ .

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Step 4.2.2.9.1

Factor $2$ out of $2 u$ .

$\frac{2 (u)}{4}$

Step 4.2.2.9.2

Cancel the common factors.

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Step 4.2.2.9.2.1

Factor $2$ out of $4$ .

$\frac{2 u}{2 \cdot 2}$

Step 4.2.2.9.2.2

Cancel the common factor.

$\frac{2 u}{2 \cdot 2}$

Step 4.2.2.9.2.3

Rewrite the expression.

$\frac{u}{2}$

Step 4.3

Reorder terms.

$\frac{1}{2} u$

Step 5

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

$\frac{u}{2}$

Step 6