Calculus Examples

$\int_{1}^{4} (\frac{x^{2}}{2} - x + 3) d x$

Step 1

Remove parentheses.

$\int_{1}^{4} \frac{x^{2}}{2} - x + 3 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{4} \frac{x^{2}}{2} d x + \int_{1}^{4} - x d x + \int_{1}^{4} 3 d x$

Step 3

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{1}^{4} x^{2} d x + \int_{1}^{4} - x d x + \int_{1}^{4} 3 d x$

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{2} \frac{1}{3} x^{3}]_{1}^{4} + \int_{1}^{4} - x d x + \int_{1}^{4} 3 d x$

Step 5

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

$\frac{1}{2} \frac{1}{3} x^{3}]_{1}^{4} - \int_{1}^{4} x d x + \int_{1}^{4} 3 d x$

Step 6

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

$\frac{1}{2} \frac{1}{3} x^{3}]_{1}^{4} - (\frac{1}{2} x^{2}]_{1}^{4}) + \int_{1}^{4} 3 d x$

Step 7

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

$\frac{1}{2} \frac{1}{3} x^{3}]_{1}^{4} - (\frac{x^{2}}{2}]_{1}^{4}) + \int_{1}^{4} 3 d x$

Step 8

Apply the constant rule.

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

Step 9

Substitute and simplify.

Tap for more steps...

Step 9.1

Evaluate $\frac{1}{3} x^{3}$ at $4$ and at $1$ .

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

Step 9.2

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

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

Step 9.3

Evaluate $3 x$ at $4$ and at $1$ .

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

Step 9.4

Simplify.

Tap for more steps...

Step 9.4.1

Raise $4$ to the power of $3$ .

$\frac{1}{2} (\frac{1}{3} \cdot 64 - \frac{1}{3} \cdot 1^{3}) - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.2

Combine $\frac{1}{3}$ and $64$ .

$\frac{1}{2} (\frac{64}{3} - \frac{1}{3} \cdot 1^{3}) - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.3

One to any power is one.

$\frac{1}{2} (\frac{64}{3} - \frac{1}{3} \cdot 1) - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.4

Multiply $- 1$ by $1$ .

$\frac{1}{2} (\frac{64}{3} - \frac{1}{3}) - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.5

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{64 - 1}{3} - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.6

Subtract $1$ from $64$ .

$\frac{1}{2} \cdot \frac{63}{3} - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.7

Cancel the common factor of $63$ and $3$ .

Tap for more steps...

Step 9.4.7.1

Factor $3$ out of $63$ .

$\frac{1}{2} \cdot \frac{3 \cdot 21}{3} - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.7.2

Cancel the common factors.

Tap for more steps...

Step 9.4.7.2.1

Factor $3$ out of $3$ .

$\frac{1}{2} \cdot \frac{3 \cdot 21}{3 (1)} - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.7.2.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{3 \cdot 21}{3 \cdot 1} - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.7.2.3

Rewrite the expression.

$\frac{1}{2} \cdot \frac{21}{1} - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.7.2.4

Divide $21$ by $1$ .

$\frac{1}{2} \cdot 21 - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.8

Combine $\frac{1}{2}$ and $21$ .

$\frac{21}{2} - (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.9

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

$\frac{21}{2} - (\frac{16}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.10

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

Tap for more steps...

Step 9.4.10.1

Factor $2$ out of $16$ .

$\frac{21}{2} - (\frac{2 \cdot 8}{2} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.10.2

Cancel the common factors.

Tap for more steps...

Step 9.4.10.2.1

Factor $2$ out of $2$ .

$\frac{21}{2} - (\frac{2 \cdot 8}{2 (1)} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.10.2.2

Cancel the common factor.

$\frac{21}{2} - (\frac{2 \cdot 8}{2 \cdot 1} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.10.2.3

Rewrite the expression.

$\frac{21}{2} - (\frac{8}{1} - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.10.2.4

Divide $8$ by $1$ .

$\frac{21}{2} - (8 - \frac{1^{2}}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.11

One to any power is one.

$\frac{21}{2} - (8 - \frac{1}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.12

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

$\frac{21}{2} - (8 \cdot \frac{2}{2} - \frac{1}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.13

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

$\frac{21}{2} - (\frac{8 \cdot 2}{2} - \frac{1}{2}) + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.14

Combine the numerators over the common denominator.

$\frac{21}{2} - \frac{8 \cdot 2 - 1}{2} + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.15

Simplify the numerator.

Tap for more steps...

Step 9.4.15.1

Multiply $8$ by $2$ .

$\frac{21}{2} - \frac{16 - 1}{2} + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.15.2

Subtract $1$ from $16$ .

$\frac{21}{2} - \frac{15}{2} + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.16

Combine the numerators over the common denominator.

$\frac{21 - 15}{2} + (3 \cdot 4) - 3 \cdot 1$

Step 9.4.17

Subtract $15$ from $21$ .

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

Step 9.4.18

Cancel the common factor of $6$ and $2$ .

Tap for more steps...

Step 9.4.18.1

Factor $2$ out of $6$ .

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

Step 9.4.18.2

Cancel the common factors.

Tap for more steps...

Step 9.4.18.2.1

Factor $2$ out of $2$ .

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

Step 9.4.18.2.2

Cancel the common factor.

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

Step 9.4.18.2.3

Rewrite the expression.

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

Step 9.4.18.2.4

Divide $3$ by $1$ .

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

Step 9.4.19

Multiply $3$ by $4$ .

$3 + 12 - 3 \cdot 1$

Step 9.4.20

Multiply $- 3$ by $1$ .

$3 + 12 - 3$

Step 9.4.21

Subtract $3$ from $12$ .

$3 + 9$

Step 9.4.22

Add $3$ and $9$ .

$12$

Step 10