Calculus Examples

$\int_{0}^{3} (x^{2} + 1) d x$

Step 1

Remove parentheses.

$\int_{0}^{3} x^{2} + 1 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{3} x^{2} d x + \int_{0}^{3} d x$

Step 3

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

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

Step 4

Apply the constant rule.

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

Step 5

Simplify the answer.

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Step 5.1

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

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

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

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

$\frac{1}{3} \cdot 27 + 3 - (\frac{1}{3} \cdot 0^{3} + 0)$

Step 5.2.2.2

Combine $\frac{1}{3}$ and $27$ .

$\frac{27}{3} + 3 - (\frac{1}{3} \cdot 0^{3} + 0)$

Step 5.2.2.3

Cancel the common factor of $27$ and $3$ .

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Step 5.2.2.3.1

Factor $3$ out of $27$ .

$\frac{3 \cdot 9}{3} + 3 - (\frac{1}{3} \cdot 0^{3} + 0)$

Step 5.2.2.3.2

Cancel the common factors.

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Step 5.2.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 9}{3 (1)} + 3 - (\frac{1}{3} \cdot 0^{3} + 0)$

Step 5.2.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 9}{3 \cdot 1} + 3 - (\frac{1}{3} \cdot 0^{3} + 0)$

Step 5.2.2.3.2.3

Rewrite the expression.

$\frac{9}{1} + 3 - (\frac{1}{3} \cdot 0^{3} + 0)$

Step 5.2.2.3.2.4

Divide $9$ by $1$ .

$9 + 3 - (\frac{1}{3} \cdot 0^{3} + 0)$

Step 5.2.2.4

Add $9$ and $3$ .

$12 - (\frac{1}{3} \cdot 0^{3} + 0)$

Step 5.2.2.5

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

$12 - (\frac{1}{3} \cdot 0 + 0)$

Step 5.2.2.6

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

$12 - (0 + 0)$

Step 5.2.2.7

Add $0$ and $0$ .

$12 - 0$

Step 5.2.2.8

Multiply $- 1$ by $0$ .

$12 + 0$

Step 5.2.2.9

Add $12$ and $0$ .

$12$

Step 6