Calculus Examples

$\int_{0}^{3} \frac{6}{9 + x^{2}} d x$

Step 1

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

$6 \int_{0}^{3} \frac{1}{9 + x^{2}} d x$

Step 2

Rewrite $9$ as $3^{2}$ .

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

Step 3

The integral of $\frac{1}{3^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{3} \arctan (\frac{x}{3})]_{0}^{3}$ .

$6 (\frac{1}{3} \arctan (\frac{x}{3})]_{0}^{3})$

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{1}{3}$ and $\arctan (\frac{x}{3})$ .

$6 (\frac{\arctan (\frac{x}{3})}{3}]_{0}^{3})$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{\arctan (\frac{x}{3})}{3}$ at $3$ and at $0$ .

$6 ((\frac{\arctan (\frac{3}{3})}{3}) - \frac{\arctan (\frac{0}{3})}{3})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$6 (\frac{\arctan (\frac{3}{3})}{3} - \frac{\arctan (\frac{0}{3})}{3})$

Step 4.2.2.1.2

Rewrite the expression.

$6 (\frac{\arctan (1)}{3} - \frac{\arctan (\frac{0}{3})}{3})$

Step 4.2.2.2

Cancel the common factor of $0$ and $3$ .

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Step 4.2.2.2.1

Factor $3$ out of $0$ .

$6 (\frac{\arctan (1)}{3} - \frac{\arctan (\frac{3 (0)}{3})}{3})$

Step 4.2.2.2.2

Cancel the common factors.

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Step 4.2.2.2.2.1

Factor $3$ out of $3$ .

$6 (\frac{\arctan (1)}{3} - \frac{\arctan (\frac{3 \cdot 0}{3 \cdot 1})}{3})$

Step 4.2.2.2.2.2

Cancel the common factor.

$6 (\frac{\arctan (1)}{3} - \frac{\arctan (\frac{3 \cdot 0}{3 \cdot 1})}{3})$

Step 4.2.2.2.2.3

Rewrite the expression.

$6 (\frac{\arctan (1)}{3} - \frac{\arctan (\frac{0}{1})}{3})$

Step 4.2.2.2.2.4

Divide $0$ by $1$ .

$6 (\frac{\arctan (1)}{3} - \frac{\arctan (0)}{3})$

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

$6 \frac{\arctan (1) - \arctan (0)}{3}$

Step 5.2

Simplify each term.

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Step 5.2.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$6 \frac{\frac{π}{4} - \arctan (0)}{3}$

Step 5.2.2

The exact value of $\arctan (0)$ is $0$ .

$6 \frac{\frac{π}{4} - 0}{3}$

Step 5.2.3

Multiply $- 1$ by $0$ .

$6 \frac{\frac{π}{4} + 0}{3}$

Step 5.3

Add $\frac{π}{4}$ and $0$ .

$6 \frac{\frac{π}{4}}{3}$

Step 5.4

Cancel the common factor of $3$ .

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Step 5.4.1

Factor $3$ out of $6$ .

$3 (2) \frac{\frac{π}{4}}{3}$

Step 5.4.2

Cancel the common factor.

$3 \cdot 2 \frac{\frac{π}{4}}{3}$

Step 5.4.3

Rewrite the expression.

$2 \frac{π}{4}$

Step 5.5

Combine $2$ and $\frac{π}{4}$ .

$\frac{2 π}{4}$

Step 5.6

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

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Step 5.6.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 5.6.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 5.6.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{2}$

Decimal Form:

$1.57079632 \dots$

Step 7