Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

$2 x]_{0}^{2} + \int_{0}^{2} x d x$

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

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

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

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $2$ by $2$ .

$4 + \frac{1}{2} \cdot 2^{2} - (2 \cdot 0 + \frac{1}{2} \cdot 0^{2})$

Step 5.2.2.2

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

$4 + \frac{1}{2} \cdot 4 - (2 \cdot 0 + \frac{1}{2} \cdot 0^{2})$

Step 5.2.2.3

Combine $\frac{1}{2}$ and $4$ .

$4 + \frac{4}{2} - (2 \cdot 0 + \frac{1}{2} \cdot 0^{2})$

Step 5.2.2.4

Cancel the common factor of $4$ and $2$ .

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Step 5.2.2.4.1

Factor $2$ out of $4$ .

$4 + \frac{2 \cdot 2}{2} - (2 \cdot 0 + \frac{1}{2} \cdot 0^{2})$

Step 5.2.2.4.2

Cancel the common factors.

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Step 5.2.2.4.2.1

Factor $2$ out of $2$ .

$4 + \frac{2 \cdot 2}{2 (1)} - (2 \cdot 0 + \frac{1}{2} \cdot 0^{2})$

Step 5.2.2.4.2.2

Cancel the common factor.

$4 + \frac{2 \cdot 2}{2 \cdot 1} - (2 \cdot 0 + \frac{1}{2} \cdot 0^{2})$

Step 5.2.2.4.2.3

Rewrite the expression.

$4 + \frac{2}{1} - (2 \cdot 0 + \frac{1}{2} \cdot 0^{2})$

Step 5.2.2.4.2.4

Divide $2$ by $1$ .

$4 + 2 - (2 \cdot 0 + \frac{1}{2} \cdot 0^{2})$

Step 5.2.2.5

Add $4$ and $2$ .

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

Step 5.2.2.6

Multiply $2$ by $0$ .

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

Step 5.2.2.7

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

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

Step 5.2.2.8

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

$6 - (0 + 0)$

Step 5.2.2.9

Add $0$ and $0$ .

$6 - 0$

Step 5.2.2.10

Multiply $- 1$ by $0$ .

$6 + 0$

Step 5.2.2.11

Add $6$ and $0$ .

$6$

Step 6