Calculus Examples

$\int_{1}^{4} 3 x d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{1}^{4} x d x$

Step 2

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

$3 (\frac{1}{2} x^{2}]_{1}^{4})$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

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

$3 (\frac{x^{2}}{2}]_{1}^{4})$

Step 3.2

Substitute and simplify.

Tap for more steps...

Step 3.2.1

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

$3 ((\frac{4^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.2.2

Simplify.

Tap for more steps...

Step 3.2.2.1

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

$3 (\frac{16}{2} - \frac{1^{2}}{2})$

Step 3.2.2.2

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

Tap for more steps...

Step 3.2.2.2.1

Factor $2$ out of $16$ .

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

Step 3.2.2.2.2

Cancel the common factors.

Tap for more steps...

Step 3.2.2.2.2.1

Factor $2$ out of $2$ .

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

Step 3.2.2.2.2.2

Cancel the common factor.

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

Step 3.2.2.2.2.3

Rewrite the expression.

$3 (\frac{8}{1} - \frac{1^{2}}{2})$

Step 3.2.2.2.2.4

Divide $8$ by $1$ .

$3 (8 - \frac{1^{2}}{2})$

Step 3.2.2.3

One to any power is one.

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

Step 3.2.2.4

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.2.2.5

Combine $8$ and $\frac{2}{2}$ .

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

Step 3.2.2.6

Combine the numerators over the common denominator.

$3 \frac{8 \cdot 2 - 1}{2}$

Step 3.2.2.7

Simplify the numerator.

Tap for more steps...

Step 3.2.2.7.1

Multiply $8$ by $2$ .

$3 \frac{16 - 1}{2}$

Step 3.2.2.7.2

Subtract $1$ from $16$ .

$3 (\frac{15}{2})$

Step 3.2.2.8

Combine $3$ and $\frac{15}{2}$ .

$\frac{3 \cdot 15}{2}$

Step 3.2.2.9

Multiply $3$ by $15$ .

$\frac{45}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{45}{2}$

Decimal Form:

$22.5$

Mixed Number Form:

$22 \frac{1}{2}$

Step 5