Calculus Examples

$\int_{1}^{5} 1 - x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{5} d x + \int_{1}^{5} - x d x$

Step 2

Apply the constant rule.

$x]_{1}^{5} + \int_{1}^{5} - x d x$

Step 3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$x]_{1}^{5} - \int_{1}^{5} x d x$

Step 4

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

$x]_{1}^{5} - (\frac{1}{2} x^{2}]_{1}^{5})$

Step 5

Simplify the answer.

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Step 5.1

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

$x]_{1}^{5} - (\frac{x^{2}}{2}]_{1}^{5})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $x$ at $5$ and at $1$ .

$(5) - 1 - (\frac{x^{2}}{2}]_{1}^{5})$

Step 5.2.2

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $1$ .

$5 - 1 - (\frac{5^{2}}{2} - \frac{1^{2}}{2})$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Subtract $1$ from $5$ .

$4 - (\frac{5^{2}}{2} - \frac{1^{2}}{2})$

Step 5.2.3.2

Raise $5$ to the power of $2$ .

$4 - (\frac{25}{2} - \frac{1^{2}}{2})$

Step 5.2.3.3

One to any power is one.

$4 - (\frac{25}{2} - \frac{1}{2})$

Step 5.2.3.4

Combine the numerators over the common denominator.

$4 - \frac{25 - 1}{2}$

Step 5.2.3.5

Subtract $1$ from $25$ .

$4 - \frac{24}{2}$

Step 5.2.3.6

Cancel the common factor of $24$ and $2$ .

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Step 5.2.3.6.1

Factor $2$ out of $24$ .

$4 - \frac{2 \cdot 12}{2}$

Step 5.2.3.6.2

Cancel the common factors.

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Step 5.2.3.6.2.1

Factor $2$ out of $2$ .

$4 - \frac{2 \cdot 12}{2 (1)}$

Step 5.2.3.6.2.2

Cancel the common factor.

$4 - \frac{2 \cdot 12}{2 \cdot 1}$

Step 5.2.3.6.2.3

Rewrite the expression.

$4 - \frac{12}{1}$

Step 5.2.3.6.2.4

Divide $12$ by $1$ .

$4 - 1 \cdot 12$

Step 5.2.3.7

Multiply $- 1$ by $12$ .

$4 - 12$

Step 5.2.3.8

Subtract $12$ from $4$ .

$- 8$

Step 6