Calculus Examples

$\int_{θ_{1}}^{θ_{2}} \frac{1}{θ_{2} - θ_{1}} x d x$

Step 1

Combine $\frac{1}{θ_{2} - θ_{1}}$ and $x$ .

$\int_{θ_{1}}^{θ_{2}} \frac{x}{θ_{2} - θ_{1}} d x$

Step 2

Since $\frac{1}{θ_{2} - θ_{1}}$ is constant with respect to $x$ , move $\frac{1}{θ_{2} - θ_{1}}$ out of the integral.

$\frac{1}{θ_{2} - θ_{1}} \int_{θ_{1}}^{θ_{2}} x d x$

Step 3

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

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

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 $x^{2}$ .

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

Step 4.1.2

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

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

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{x^{2}}{2}$ at $θ_{2}$ and at $θ_{1}$ .

$\frac{(\frac{{θ_{2}}^{2}}{2}) - \frac{{θ_{1}}^{2}}{2}}{θ_{2} - θ_{1}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Multiply $\frac{\frac{{θ_{2}}^{2}}{2} - \frac{{θ_{1}}^{2}}{2}}{θ_{2} - θ_{1}}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{\frac{{θ_{2}}^{2}}{2} - \frac{{θ_{1}}^{2}}{2}}{θ_{2} - θ_{1}}$

Step 4.2.2.2

Combine.

$\frac{2 (\frac{{θ_{2}}^{2}}{2} - \frac{{θ_{1}}^{2}}{2})}{2 (θ_{2} - θ_{1})}$

Step 4.2.2.3

Apply the distributive property.

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

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.

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

Step 4.2.2.4.2

Rewrite the expression.

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

Step 4.2.2.5

Multiply $- 1$ by $2$ .

$\frac{{θ_{2}}^{2} - 2 \frac{{θ_{1}}^{2}}{2}}{2 (θ_{2} - θ_{1})}$

Step 4.2.2.6

Combine $- 2$ and $\frac{{θ_{1}}^{2}}{2}$ .

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

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 $- 2 {θ_{1}}^{2}$ .

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

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 $2$ .

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

Step 4.2.2.7.2.2

Cancel the common factor.

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

Step 4.2.2.7.2.3

Rewrite the expression.

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

Step 4.2.2.7.2.4

Divide $- {θ_{1}}^{2}$ by $1$ .

$\frac{{θ_{2}}^{2} - {θ_{1}}^{2}}{2 (θ_{2} - θ_{1})}$

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 $a = θ_{2}$ and $b = θ_{1}$ .

$\frac{(θ_{2} + θ_{1}) (θ_{2} - θ_{1})}{2 (θ_{2} - θ_{1})}$

Step 5.2

Cancel the common factor of $θ_{2} - θ_{1}$ .

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Step 5.2.1

Cancel the common factor.

$\frac{(θ_{2} + θ_{1}) (θ_{2} - θ_{1})}{2 (θ_{2} - θ_{1})}$

Step 5.2.2

Rewrite the expression.

$\frac{θ_{2} + θ_{1}}{2}$