Calculus Examples

$\int_{0}^{6} (x^{2} + x + 1) d x$

Step 1

Remove parentheses.

$\int_{0}^{6} x^{2} + x + 1 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{6} x^{2} d x + \int_{0}^{6} x d x + \int_{0}^{6} d x$

Step 3

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

$\frac{1}{3} x^{3}]_{0}^{6} + \int_{0}^{6} x d x + \int_{0}^{6} d x$

Step 4

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

$\frac{1}{3} x^{3}]_{0}^{6} + \frac{1}{2} x^{2}]_{0}^{6} + \int_{0}^{6} d x$

Step 5

Apply the constant rule.

$\frac{1}{3} x^{3}]_{0}^{6} + \frac{1}{2} x^{2}]_{0}^{6} + x]_{0}^{6}$

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Simplify.

Tap for more steps...

Step 6.1.1

Combine $\frac{1}{3} x^{3}]_{0}^{6}$ and $\frac{1}{2} x^{2}]_{0}^{6}$ .

$\frac{1}{3} x^{3} + \frac{1}{2} x^{2}]_{0}^{6} + x]_{0}^{6}$

Step 6.1.2

Combine $\frac{1}{3} x^{3} + \frac{1}{2} x^{2}]_{0}^{6}$ and $x]_{0}^{6}$ .

$\frac{1}{3} x^{3} + \frac{1}{2} x^{2} + x]_{0}^{6}$

Step 6.2

Substitute and simplify.

Tap for more steps...

Step 6.2.1

Evaluate $\frac{1}{3} x^{3} + \frac{1}{2} x^{2} + x$ at $6$ and at $0$ .

$(\frac{1}{3} \cdot 6^{3} + \frac{1}{2} \cdot 6^{2} + 6) - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2

Simplify.

Tap for more steps...

Step 6.2.2.1

Raise $6$ to the power of $3$ .

$\frac{1}{3} \cdot 216 + \frac{1}{2} \cdot 6^{2} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.2

Combine $\frac{1}{3}$ and $216$ .

$\frac{216}{3} + \frac{1}{2} \cdot 6^{2} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.3

Cancel the common factor of $216$ and $3$ .

Tap for more steps...

Step 6.2.2.3.1

Factor $3$ out of $216$ .

$\frac{3 \cdot 72}{3} + \frac{1}{2} \cdot 6^{2} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.3.2

Cancel the common factors.

Tap for more steps...

Step 6.2.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 72}{3 (1)} + \frac{1}{2} \cdot 6^{2} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 72}{3 \cdot 1} + \frac{1}{2} \cdot 6^{2} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.3.2.3

Rewrite the expression.

$\frac{72}{1} + \frac{1}{2} \cdot 6^{2} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.3.2.4

Divide $72$ by $1$ .

$72 + \frac{1}{2} \cdot 6^{2} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.4

Raise $6$ to the power of $2$ .

$72 + \frac{1}{2} \cdot 36 + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.5

Combine $\frac{1}{2}$ and $36$ .

$72 + \frac{36}{2} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.6

Cancel the common factor of $36$ and $2$ .

Tap for more steps...

Step 6.2.2.6.1

Factor $2$ out of $36$ .

$72 + \frac{2 \cdot 18}{2} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.6.2

Cancel the common factors.

Tap for more steps...

Step 6.2.2.6.2.1

Factor $2$ out of $2$ .

$72 + \frac{2 \cdot 18}{2 (1)} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.6.2.2

Cancel the common factor.

$72 + \frac{2 \cdot 18}{2 \cdot 1} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.6.2.3

Rewrite the expression.

$72 + \frac{18}{1} + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.6.2.4

Divide $18$ by $1$ .

$72 + 18 + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.7

Add $72$ and $18$ .

$90 + 6 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.8

Add $90$ and $6$ .

$96 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.9

Raising $0$ to any positive power yields $0$ .

$96 - (\frac{1}{3} \cdot 0 + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.10

Multiply $\frac{1}{3}$ by $0$ .

$96 - (0 + \frac{1}{2} \cdot 0^{2} + 0)$

Step 6.2.2.11

Raising $0$ to any positive power yields $0$ .

$96 - (0 + \frac{1}{2} \cdot 0 + 0)$

Step 6.2.2.12

Multiply $\frac{1}{2}$ by $0$ .

$96 - (0 + 0 + 0)$

Step 6.2.2.13

Add $0$ and $0$ .

$96 - (0 + 0)$

Step 6.2.2.14

Add $0$ and $0$ .

$96 - 0$

Step 6.2.2.15

Multiply $- 1$ by $0$ .

$96 + 0$

Step 6.2.2.16

Add $96$ and $0$ .

$96$

Step 7