Calculus Examples

$\int_{1}^{4} 6 x - 5 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{4} 6 x d x + \int_{1}^{4} - 5 d x$

Step 2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int_{1}^{4} x d x + \int_{1}^{4} - 5 d x$

Step 3

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

$6 (\frac{1}{2} x^{2}]_{1}^{4}) + \int_{1}^{4} - 5 d x$

Step 4

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

$6 (\frac{x^{2}}{2}]_{1}^{4}) + \int_{1}^{4} - 5 d x$

Step 5

Apply the constant rule.

$6 (\frac{x^{2}}{2}]_{1}^{4}) + - 5 x]_{1}^{4}$

Step 6

Substitute and simplify.

Tap for more steps...

Step 6.1

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

$6 ((\frac{4^{2}}{2}) - \frac{1^{2}}{2}) + - 5 x]_{1}^{4}$

Step 6.2

Evaluate $- 5 x$ at $4$ and at $1$ .

$6 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + - 5 \cdot 4 + 5 \cdot 1$

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

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

$6 (\frac{16}{2} - \frac{1^{2}}{2}) - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.2

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

Tap for more steps...

Step 6.3.2.1

Factor $2$ out of $16$ .

$6 (\frac{2 \cdot 8}{2} - \frac{1^{2}}{2}) - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.2.2

Cancel the common factors.

Tap for more steps...

Step 6.3.2.2.1

Factor $2$ out of $2$ .

$6 (\frac{2 \cdot 8}{2 (1)} - \frac{1^{2}}{2}) - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.2.2.2

Cancel the common factor.

$6 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{1^{2}}{2}) - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.2.2.3

Rewrite the expression.

$6 (\frac{8}{1} - \frac{1^{2}}{2}) - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.2.2.4

Divide $8$ by $1$ .

$6 (8 - \frac{1^{2}}{2}) - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.3

One to any power is one.

$6 (8 - \frac{1}{2}) - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.4

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

$6 (8 \cdot \frac{2}{2} - \frac{1}{2}) - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.5

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

$6 (\frac{8 \cdot 2}{2} - \frac{1}{2}) - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.6

Combine the numerators over the common denominator.

$6 \frac{8 \cdot 2 - 1}{2} - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.7

Simplify the numerator.

Tap for more steps...

Step 6.3.7.1

Multiply $8$ by $2$ .

$6 \frac{16 - 1}{2} - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.7.2

Subtract $1$ from $16$ .

$6 (\frac{15}{2}) - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.8

Combine $6$ and $\frac{15}{2}$ .

$\frac{6 \cdot 15}{2} - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.9

Multiply $6$ by $15$ .

$\frac{90}{2} - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.10

Cancel the common factor of $90$ and $2$ .

Tap for more steps...

Step 6.3.10.1

Factor $2$ out of $90$ .

$\frac{2 \cdot 45}{2} - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.10.2

Cancel the common factors.

Tap for more steps...

Step 6.3.10.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 45}{2 (1)} - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.10.2.2

Cancel the common factor.

$\frac{2 \cdot 45}{2 \cdot 1} - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.10.2.3

Rewrite the expression.

$\frac{45}{1} - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.10.2.4

Divide $45$ by $1$ .

$45 - 5 \cdot 4 + 5 \cdot 1$

Step 6.3.11

Multiply $- 5$ by $4$ .

$45 - 20 + 5 \cdot 1$

Step 6.3.12

Multiply $5$ by $1$ .

$45 - 20 + 5$

Step 6.3.13

Add $- 20$ and $5$ .

$45 - 15$

Step 6.3.14

Subtract $15$ from $45$ .

$30$

Step 7