Calculus Examples

$\int_{a}^{b} \frac{1}{b - a} x d x$

Step 1

Combine $\frac{1}{b - a}$ and $x$ .

$\int_{a}^{b} \frac{x}{b - a} d x$

Step 2

Since $\frac{1}{b - a}$ is constant with respect to $x$ , move $\frac{1}{b - a}$ out of the integral.

$\frac{1}{b - a} \int_{a}^{b} x d x$

Step 3

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

$\frac{1}{b - a} \frac{1}{2} x^{2}]_{a}^{b}$

Step 4

Simplify the answer.

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Step 4.1

Simplify.

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Step 4.1.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{1}{b - a} \frac{x^{2}}{2}]_{a}^{b}$

Step 4.1.2

Combine $\frac{1}{b - a}$ and $\frac{x^{2}}{2}]_{a}^{b}$ .

$\frac{\frac{x^{2}}{2}]_{a}^{b}}{b - a}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{x^{2}}{2}$ at $b$ and at $a$ .

$\frac{(\frac{b^{2}}{2}) - \frac{a^{2}}{2}}{b - a}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Multiply $\frac{\frac{b^{2}}{2} - \frac{a^{2}}{2}}{b - a}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{\frac{b^{2}}{2} - \frac{a^{2}}{2}}{b - a}$

Step 4.2.2.2

Combine.

$\frac{2 (\frac{b^{2}}{2} - \frac{a^{2}}{2})}{2 (b - a)}$

Step 4.2.2.3

Apply the distributive property.

$\frac{2 \frac{b^{2}}{2} + 2 (- \frac{a^{2}}{2})}{2 (b - a)}$

Step 4.2.2.4

Cancel the common factor of $2$ .

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Step 4.2.2.4.1

Cancel the common factor.

$\frac{2 \frac{b^{2}}{2} + 2 (- \frac{a^{2}}{2})}{2 (b - a)}$

Step 4.2.2.4.2

Rewrite the expression.

$\frac{b^{2} + 2 (- \frac{a^{2}}{2})}{2 (b - a)}$

Step 4.2.2.5

Multiply $- 1$ by $2$ .

$\frac{b^{2} - 2 \frac{a^{2}}{2}}{2 (b - a)}$

Step 4.2.2.6

Combine $- 2$ and $\frac{a^{2}}{2}$ .

$\frac{b^{2} + \frac{- 2 a^{2}}{2}}{2 (b - a)}$

Step 4.2.2.7

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

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Step 4.2.2.7.1

Factor $2$ out of $- 2 a^{2}$ .

$\frac{b^{2} + \frac{2 (- a^{2})}{2}}{2 (b - a)}$

Step 4.2.2.7.2

Cancel the common factors.

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Step 4.2.2.7.2.1

Factor $2$ out of $2$ .

$\frac{b^{2} + \frac{2 (- a^{2})}{2 (1)}}{2 (b - a)}$

Step 4.2.2.7.2.2

Cancel the common factor.

$\frac{b^{2} + \frac{2 (- a^{2})}{2 \cdot 1}}{2 (b - a)}$

Step 4.2.2.7.2.3

Rewrite the expression.

$\frac{b^{2} + \frac{- a^{2}}{1}}{2 (b - a)}$

Step 4.2.2.7.2.4

Divide $- a^{2}$ by $1$ .

$\frac{b^{2} - a^{2}}{2 (b - a)}$

Step 5

Simplify.

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Step 5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = b$ and $b = a$ .

$\frac{(b + a) (b - a)}{2 (b - a)}$

Step 5.2

Cancel the common factor of $b - a$ .

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Step 5.2.1

Cancel the common factor.

$\frac{(b + a) (b - a)}{2 (b - a)}$

Step 5.2.2

Rewrite the expression.

$\frac{b + a}{2}$