Calculus Examples

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

Step 1

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

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

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $\frac{1}{2} x^{2}$ at $2$ and at $4$ .

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

Step 2.2

Simplify.

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Step 2.2.1

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

$\frac{1}{2} \cdot 4 - \frac{1}{2} \cdot 4^{2}$

Step 2.2.2

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

$\frac{4}{2} - \frac{1}{2} \cdot 4^{2}$

Step 2.2.3

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

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Step 2.2.3.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2} - \frac{1}{2} \cdot 4^{2}$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $2$ out of $2$ .

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

Step 2.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} - \frac{1}{2} \cdot 4^{2}$

Step 2.2.3.2.3

Rewrite the expression.

$\frac{2}{1} - \frac{1}{2} \cdot 4^{2}$

Step 2.2.3.2.4

Divide $2$ by $1$ .

$2 - \frac{1}{2} \cdot 4^{2}$

Step 2.2.4

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

$2 - \frac{1}{2} \cdot 16$

Step 2.2.5

Multiply $16$ by $- 1$ .

$2 - 16 (\frac{1}{2})$

Step 2.2.6

Combine $- 16$ and $\frac{1}{2}$ .

$2 + \frac{- 16}{2}$

Step 2.2.7

Cancel the common factor of $- 16$ and $2$ .

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Step 2.2.7.1

Factor $2$ out of $- 16$ .

$2 + \frac{2 \cdot - 8}{2}$

Step 2.2.7.2

Cancel the common factors.

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Step 2.2.7.2.1

Factor $2$ out of $2$ .

$2 + \frac{2 \cdot - 8}{2 (1)}$

Step 2.2.7.2.2

Cancel the common factor.

$2 + \frac{2 \cdot - 8}{2 \cdot 1}$

Step 2.2.7.2.3

Rewrite the expression.

$2 + \frac{- 8}{1}$

Step 2.2.7.2.4

Divide $- 8$ by $1$ .

$2 - 8$

Step 2.2.8

Subtract $8$ from $2$ .

$- 6$

Step 3