Calculus Examples

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

Since integration is linear, the integral of $6 x - 2$ with respect to $x$ is $\int_{1}^{3} 6 x d x + \int_{1}^{3} - 2 d x$ .

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

Since $6$ is constant with respect to $x$ , the integral of $6 x$ with respect to $x$ is $6 \int_{1}^{3} x d x$ .

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

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

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

Combine fractions.

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Write $x^{2}$ as a fraction with denominator $1$ .

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

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

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

Since $- 2$ is constant with respect to $x$ , the integral of $- 2$ with respect to $x$ is $- 2 x$ .

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

Simplify the answer.

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Evaluate $\frac{x^{2}}{2}$ at $3$ and at $1$ .

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

Evaluate $- 2 x$ at $3$ and at $1$ .

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

Raise $3$ to the power of $2$ .

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

One to any power is one.

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 1$ by $1$ .

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

Subtract $1$ from $9$ .

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

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

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

Multiply $6$ by $4$ .

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

Multiply $- 2$ by $3$ .

$24 + (- 6 + 2 \cdot 1)$

Multiply $2$ by $1$ .

$24 + (- 6 + 2)$

Add $- 6$ and $2$ .

$24 - 4$

Subtract $4$ from $24$ .

$20$

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