Calculus Examples

$\int_{1}^{3} 6 x - 2 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{3} 6 x d x + \int_{1}^{3} - 2 d x$

Step 2

Since

$6$ is constant with respect to

$x$ , move

$6$ out of the integral.

$6 \int_{1}^{3} x d x + \int_{1}^{3} - 2 d x$

Step 3

By the Power Rule, the integral of

$x$ with respect to

$x$ is

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

$6 (\frac{1}{2} x^{2}]_{1}^{3}) + \int_{1}^{3} - 2 d x$

Step 4

Combine

$\frac{1}{2}$ and

$x^{2}$ .

$6 (\frac{x^{2}}{2}]_{1}^{3}) + \int_{1}^{3} - 2 d x$

Step 5

Apply the constant rule.

$6 (\frac{x^{2}}{2}]_{1}^{3}) + - 2 x]_{1}^{3}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate

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

$3$ and at

$1$ .

$6 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2}) + - 2 x]_{1}^{3}$

Step 6.2

Evaluate

$- 2 x$ at

$3$ and at

$1$ .

$6 (\frac{3^{2}}{2} - \frac{1^{2}}{2}) + - 2 \cdot 3 + 2 \cdot 1$

Step 6.3

Simplify.

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Step 6.3.1

Raise

$3$ to the power of

$2$ .

$6 (\frac{9}{2} - \frac{1^{2}}{2}) - 2 \cdot 3 + 2 \cdot 1$

Step 6.3.2

One to any power is one.

$6 (\frac{9}{2} - \frac{1}{2}) - 2 \cdot 3 + 2 \cdot 1$

Step 6.3.3

Combine the numerators over the common denominator.

$6 \frac{9 - 1}{2} - 2 \cdot 3 + 2 \cdot 1$

Step 6.3.4

Subtract

$1$ from

$9$ .

$6 (\frac{8}{2}) - 2 \cdot 3 + 2 \cdot 1$

Step 6.3.5

Cancel the common factor of

$8$ and

$2$ .

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Step 6.3.5.1

Factor

$2$ out of

$8$ .

$6 \frac{2 \cdot 4}{2} - 2 \cdot 3 + 2 \cdot 1$

Step 6.3.5.2

Cancel the common factors.

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Step 6.3.5.2.1

Factor

$2$ out of

$2$ .

$6 \frac{2 \cdot 4}{2 (1)} - 2 \cdot 3 + 2 \cdot 1$

Step 6.3.5.2.2

Cancel the common factor.

$6 \frac{2 \cdot 4}{2 \cdot 1} - 2 \cdot 3 + 2 \cdot 1$

Step 6.3.5.2.3

Rewrite the expression.

$6 (\frac{4}{1}) - 2 \cdot 3 + 2 \cdot 1$

Step 6.3.5.2.4

Divide

$4$ by

$1$ .

$6 \cdot 4 - 2 \cdot 3 + 2 \cdot 1$

Step 6.3.6

Multiply

$6$ by

$4$ .

$24 - 2 \cdot 3 + 2 \cdot 1$

Step 6.3.7

Multiply

$- 2$ by

$3$ .

$24 - 6 + 2 \cdot 1$

Step 6.3.8

Multiply

$2$ by

$1$ .

$24 - 6 + 2$

Step 6.3.9

Add

$- 6$ and

$2$ .

$24 - 4$

Step 6.3.10

Subtract

$4$ from

$24$ .

$20$

Step 7

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