Calculus Examples

$\int_{2}^{4} d x \int_{x}^{2 x} \frac{y}{3 x} d y$

Step 1

Evaluate $\int_{2}^{4} d x$ .

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Step 1.1

Apply the constant rule.

$x]_{2}^{4} \int_{x}^{2 x} \frac{y}{3 x} d y$

Step 1.2

Substitute and simplify.

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Step 1.2.1

Evaluate $x$ at $4$ and at $2$ .

$((4) - 2) \int_{x}^{2 x} \frac{y}{3 x} d y$

Step 1.2.2

Subtract $2$ from $4$ .

$2 \int_{x}^{2 x} \frac{y}{3 x} d y$

Step 2

Evaluate $\int_{x}^{2 x} \frac{y}{3 x} d y$ .

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Step 2.1

Since $\frac{1}{3 x}$ is constant with respect to $y$ , move $\frac{1}{3 x}$ out of the integral.

$2 (\frac{1}{3 x} \int_{x}^{2 x} y d y)$

Step 2.2

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

$2 (\frac{1}{3 x} \frac{1}{2} y^{2}]_{x}^{2 x})$

Step 2.3

Simplify the answer.

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Step 2.3.1

Simplify.

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Step 2.3.1.1

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

$2 (\frac{1}{3 x} \frac{y^{2}}{2}]_{x}^{2 x})$

Step 2.3.1.2

Combine $\frac{1}{3 x}$ and $\frac{y^{2}}{2}]_{x}^{2 x}$ .

$2 \frac{\frac{y^{2}}{2}]_{x}^{2 x}}{3 x}$

Step 2.3.2

Substitute and simplify.

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Step 2.3.2.1

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

$2 \frac{(\frac{{(2 x)}^{2}}{2}) - \frac{x^{2}}{2}}{3 x}$

Step 2.3.2.2

Simplify.

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Step 2.3.2.2.1

Factor $2$ out of $2 x$ .

$2 \frac{\frac{{(2 (x))}^{2}}{2} - \frac{x^{2}}{2}}{3 x}$

Step 2.3.2.2.2

Apply the product rule to $2 (x)$ .

$2 \frac{\frac{2^{2} x^{2}}{2} - \frac{x^{2}}{2}}{3 x}$

Step 2.3.2.2.3

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

$2 \frac{\frac{4 x^{2}}{2} - \frac{x^{2}}{2}}{3 x}$

Step 2.3.2.2.4

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

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Step 2.3.2.2.4.1

Factor $2$ out of $4 x^{2}$ .

$2 \frac{\frac{2 (2 x^{2})}{2} - \frac{x^{2}}{2}}{3 x}$

Step 2.3.2.2.4.2

Cancel the common factors.

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Step 2.3.2.2.4.2.1

Factor $2$ out of $2$ .

$2 \frac{\frac{2 (2 x^{2})}{2 (1)} - \frac{x^{2}}{2}}{3 x}$

Step 2.3.2.2.4.2.2

Cancel the common factor.

$2 \frac{\frac{2 (2 x^{2})}{2 \cdot 1} - \frac{x^{2}}{2}}{3 x}$

Step 2.3.2.2.4.2.3

Rewrite the expression.

$2 \frac{\frac{2 x^{2}}{1} - \frac{x^{2}}{2}}{3 x}$

Step 2.3.2.2.4.2.4

Divide $2 x^{2}$ by $1$ .

$2 \frac{2 x^{2} - \frac{x^{2}}{2}}{3 x}$

Step 2.3.2.2.5

To write $2 x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 \frac{2 x^{2} \cdot \frac{2}{2} - \frac{x^{2}}{2}}{3 x}$

Step 2.3.2.2.6

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

$2 \frac{\frac{2 x^{2} \cdot 2}{2} - \frac{x^{2}}{2}}{3 x}$

Step 2.3.2.2.7

Combine the numerators over the common denominator.

$2 \frac{\frac{2 x^{2} \cdot 2 - x^{2}}{2}}{3 x}$

Step 2.3.2.2.8

Multiply $2$ by $2$ .

$2 \frac{\frac{4 x^{2} - x^{2}}{2}}{3 x}$

Step 2.3.2.2.9

Subtract $x^{2}$ from $4 x^{2}$ .

$2 \frac{\frac{3 x^{2}}{2}}{3 x}$

Step 2.3.2.2.10

Rewrite $\frac{\frac{3 x^{2}}{2}}{3 x}$ as a product.

$2 (\frac{3 x^{2}}{2} \cdot \frac{1}{3 x})$

Step 2.3.2.2.11

Multiply $\frac{3 x^{2}}{2}$ by $\frac{1}{3 x}$ .

$2 \frac{3 x^{2}}{2 (3 x)}$

Step 2.3.2.2.12

Multiply $3$ by $2$ .

$2 \frac{3 x^{2}}{6 x}$

Step 2.3.2.2.13

Cancel the common factor of $3$ and $6$ .

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Step 2.3.2.2.13.1

Factor $3$ out of $3 x^{2}$ .

$2 \frac{3 (x^{2})}{6 x}$

Step 2.3.2.2.13.2

Cancel the common factors.

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Step 2.3.2.2.13.2.1

Factor $3$ out of $6 x$ .

$2 \frac{3 (x^{2})}{3 (2 x)}$

Step 2.3.2.2.13.2.2

Cancel the common factor.

$2 \frac{3 x^{2}}{3 (2 x)}$

Step 2.3.2.2.13.2.3

Rewrite the expression.

$2 \frac{x^{2}}{2 x}$

Step 2.3.2.2.14

Cancel the common factor of $x^{2}$ and $x$ .

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Step 2.3.2.2.14.1

Factor $x$ out of $x^{2}$ .

$2 \frac{x \cdot x}{2 x}$

Step 2.3.2.2.14.2

Cancel the common factors.

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Step 2.3.2.2.14.2.1

Factor $x$ out of $2 x$ .

$2 \frac{x \cdot x}{x \cdot 2}$

Step 2.3.2.2.14.2.2

Cancel the common factor.

$2 \frac{x \cdot x}{x \cdot 2}$

Step 2.3.2.2.14.2.3

Rewrite the expression.

$2 \frac{x}{2}$

Step 2.3.3

Reorder terms.

$2 (\frac{1}{2} x)$

Step 2.4

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

$2 \frac{x}{2}$

Step 3

Simplify.

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Step 3.1

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

$\frac{2 x}{2}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{2 x}{2}$

Step 3.2.2

Divide $x$ by $1$ .

$x$