Calculus Examples

$\int_{0}^{\infty} \frac{1}{4 x^{2} + 9} d x$

Step 1

Write the integral as a limit as $t$ approaches $\infty$ .

$lim_{t \to \infty} \int_{0}^{t} \frac{1}{4 x^{2} + 9} d x$

Step 2

Reorder $4 x^{2}$ and $9$ .

$lim_{t \to \infty} \int_{0}^{t} \frac{1}{9 + 4 x^{2}} d x$

Step 3

Factor $4$ out of $9 + 4 x^{2}$ .

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Step 3.1

Factor $4$ out of $9$ .

$lim_{t \to \infty} \int_{0}^{t} \frac{1}{4 (\frac{9}{4}) + 4 x^{2}} d x$

Step 3.2

Factor $4$ out of $4 x^{2}$ .

$lim_{t \to \infty} \int_{0}^{t} \frac{1}{4 (\frac{9}{4}) + 4 (x^{2})} d x$

Step 3.3

Factor $4$ out of $4 (\frac{9}{4}) + 4 (x^{2})$ .

$lim_{t \to \infty} \int_{0}^{t} \frac{1}{4 (\frac{9}{4} + x^{2})} d x$

Step 4

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$lim_{t \to \infty} \frac{1}{4} \int_{0}^{t} \frac{1}{\frac{9}{4} + x^{2}} d x$

Step 5

Rewrite $\frac{9}{4}$ as ${(\frac{3}{2})}^{2}$ .

$lim_{t \to \infty} \frac{1}{4} \int_{0}^{t} \frac{1}{{(\frac{3}{2})}^{2} + x^{2}} d x$

Step 6

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

$lim_{t \to \infty} \frac{1}{4} \frac{1}{\frac{3}{2}} \arctan (\frac{x}{\frac{3}{2}})]_{0}^{t}$

Step 7

Simplify the answer.

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Step 7.1

Simplify.

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Step 7.1.1

Multiply by the reciprocal of the fraction to divide by $\frac{3}{2}$ .

$lim_{t \to \infty} \frac{1}{4} 1 (\frac{2}{3}) \arctan (\frac{x}{\frac{3}{2}})]_{0}^{t}$

Step 7.1.2

Multiply $\frac{2}{3}$ by $1$ .

$lim_{t \to \infty} \frac{1}{4} \frac{2}{3} \arctan (\frac{x}{\frac{3}{2}})]_{0}^{t}$

Step 7.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{3}{2}$ .

$lim_{t \to \infty} \frac{1}{4} \frac{2}{3} \arctan (x \frac{2}{3})]_{0}^{t}$

Step 7.1.4

Combine $x$ and $\frac{2}{3}$ .

$lim_{t \to \infty} \frac{1}{4} \frac{2}{3} \arctan (\frac{x \cdot 2}{3})]_{0}^{t}$

Step 7.1.5

Move $2$ to the left of $x$ .

$lim_{t \to \infty} \frac{1}{4} \frac{2}{3} \arctan (\frac{2 \cdot x}{3})]_{0}^{t}$

Step 7.1.6

Combine $\frac{2}{3}$ and $\arctan (\frac{2 x}{3})$ .

$lim_{t \to \infty} \frac{1}{4} \frac{2 \arctan (\frac{2 x}{3})}{3}]_{0}^{t}$

Step 7.1.7

Combine $\frac{1}{4}$ and $\frac{2 \arctan (\frac{2 x}{3})}{3}]_{0}^{t}$ .

$lim_{t \to \infty} \frac{\frac{2 \arctan (\frac{2 x}{3})}{3}]_{0}^{t}}{4}$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{2 \arctan (\frac{2 x}{3})}{3}$ at $t$ and at $0$ .

$lim_{t \to \infty} \frac{(\frac{2 \arctan (\frac{2 t}{3})}{3}) - \frac{2 \arctan (\frac{2 \cdot 0}{3})}{3}}{4}$

Step 7.2.2

Simplify.

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Step 7.2.2.1

Multiply $2$ by $0$ .

$lim_{t \to \infty} \frac{\frac{2 \arctan (\frac{2 t}{3})}{3} - \frac{2 \arctan (\frac{0}{3})}{3}}{4}$

Step 7.2.2.2

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

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Step 7.2.2.2.1

Factor $3$ out of $0$ .

$lim_{t \to \infty} \frac{\frac{2 \arctan (\frac{2 t}{3})}{3} - \frac{2 \arctan (\frac{3 (0)}{3})}{3}}{4}$

Step 7.2.2.2.2

Cancel the common factors.

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Step 7.2.2.2.2.1

Factor $3$ out of $3$ .

$lim_{t \to \infty} \frac{\frac{2 \arctan (\frac{2 t}{3})}{3} - \frac{2 \arctan (\frac{3 \cdot 0}{3 \cdot 1})}{3}}{4}$

Step 7.2.2.2.2.2

Cancel the common factor.

$lim_{t \to \infty} \frac{\frac{2 \arctan (\frac{2 t}{3})}{3} - \frac{2 \arctan (\frac{3 \cdot 0}{3 \cdot 1})}{3}}{4}$

Step 7.2.2.2.2.3

Rewrite the expression.

$lim_{t \to \infty} \frac{\frac{2 \arctan (\frac{2 t}{3})}{3} - \frac{2 \arctan (\frac{0}{1})}{3}}{4}$

Step 7.2.2.2.2.4

Divide $0$ by $1$ .

$lim_{t \to \infty} \frac{\frac{2 \arctan (\frac{2 t}{3})}{3} - \frac{2 \arctan (0)}{3}}{4}$

Step 7.2.2.3

Multiply $\frac{\frac{2 \arctan (\frac{2 t}{3})}{3} - \frac{2 \arctan (0)}{3}}{4}$ by $\frac{3}{3}$ .

$lim_{t \to \infty} \frac{3}{3} \cdot \frac{\frac{2 \arctan (\frac{2 t}{3})}{3} - \frac{2 \arctan (0)}{3}}{4}$

Step 7.2.2.4

Combine.

$lim_{t \to \infty} \frac{3 (\frac{2 \arctan (\frac{2 t}{3})}{3} - \frac{2 \arctan (0)}{3})}{3 \cdot 4}$

Step 7.2.2.5

Apply the distributive property.

$lim_{t \to \infty} \frac{3 \frac{2 \arctan (\frac{2 t}{3})}{3} + 3 (- \frac{2 \arctan (0)}{3})}{3 \cdot 4}$

Step 7.2.2.6

Cancel the common factor of $3$ .

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Step 7.2.2.6.1

Cancel the common factor.

$lim_{t \to \infty} \frac{3 \frac{2 \arctan (\frac{2 t}{3})}{3} + 3 (- \frac{2 \arctan (0)}{3})}{3 \cdot 4}$

Step 7.2.2.6.2

Rewrite the expression.

$lim_{t \to \infty} \frac{2 \arctan (\frac{2 t}{3}) + 3 (- \frac{2 \arctan (0)}{3})}{3 \cdot 4}$

Step 7.2.2.7

Multiply $- 1$ by $3$ .

$lim_{t \to \infty} \frac{2 \arctan (\frac{2 t}{3}) - 3 \frac{2 \arctan (0)}{3}}{3 \cdot 4}$

Step 7.2.2.8

Combine $- 3$ and $\frac{2 \arctan (0)}{3}$ .

$lim_{t \to \infty} \frac{2 \arctan (\frac{2 t}{3}) + \frac{- 3 (2 \arctan (0))}{3}}{3 \cdot 4}$

Step 7.2.2.9

Multiply $2$ by $- 3$ .

$lim_{t \to \infty} \frac{2 \arctan (\frac{2 t}{3}) + \frac{- 6 \arctan (0)}{3}}{3 \cdot 4}$

Step 7.2.2.10

Cancel the common factor of $- 6$ and $3$ .

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Step 7.2.2.10.1

Factor $3$ out of $- 6 \arctan (0)$ .

$lim_{t \to \infty} \frac{2 \arctan (\frac{2 t}{3}) + \frac{3 (- 2 \arctan (0))}{3}}{3 \cdot 4}$

Step 7.2.2.10.2

Cancel the common factors.

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Step 7.2.2.10.2.1

Factor $3$ out of $3$ .

$lim_{t \to \infty} \frac{2 \arctan (\frac{2 t}{3}) + \frac{3 (- 2 \arctan (0))}{3 (1)}}{3 \cdot 4}$

Step 7.2.2.10.2.2

Cancel the common factor.

$lim_{t \to \infty} \frac{2 \arctan (\frac{2 t}{3}) + \frac{3 (- 2 \arctan (0))}{3 \cdot 1}}{3 \cdot 4}$

Step 7.2.2.10.2.3

Rewrite the expression.

$lim_{t \to \infty} \frac{2 \arctan (\frac{2 t}{3}) + \frac{- 2 \arctan (0)}{1}}{3 \cdot 4}$

Step 7.2.2.10.2.4

Divide $- 2 \arctan (0)$ by $1$ .

$lim_{t \to \infty} \frac{2 \arctan (\frac{2 t}{3}) - 2 \arctan (0)}{3 \cdot 4}$

Step 7.2.2.11

Multiply $3$ by $4$ .

$lim_{t \to \infty} \frac{2 \arctan (\frac{2 t}{3}) - 2 \arctan (0)}{12}$

Step 7.2.2.12

Cancel the common factor of $2 \arctan (\frac{2 t}{3}) - 2 \arctan (0)$ and $12$ .

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Step 7.2.2.12.1

Factor $2$ out of $2 \arctan (\frac{2 t}{3})$ .

$lim_{t \to \infty} \frac{2 (\arctan (\frac{2 t}{3})) - 2 \arctan (0)}{12}$

Step 7.2.2.12.2

Factor $2$ out of $- 2 \arctan (0)$ .

$lim_{t \to \infty} \frac{2 (\arctan (\frac{2 t}{3})) + 2 (- \arctan (0))}{12}$

Step 7.2.2.12.3

Factor $2$ out of $2 (\arctan (\frac{2 t}{3})) + 2 (- \arctan (0))$ .

$lim_{t \to \infty} \frac{2 (\arctan (\frac{2 t}{3}) - \arctan (0))}{12}$

Step 7.2.2.12.4

Cancel the common factors.

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Step 7.2.2.12.4.1

Factor $2$ out of $12$ .

$lim_{t \to \infty} \frac{2 (\arctan (\frac{2 t}{3}) - \arctan (0))}{2 (6)}$

Step 7.2.2.12.4.2

Cancel the common factor.

$lim_{t \to \infty} \frac{2 (\arctan (\frac{2 t}{3}) - \arctan (0))}{2 \cdot 6}$

Step 7.2.2.12.4.3

Rewrite the expression.

$lim_{t \to \infty} \frac{\arctan (\frac{2 t}{3}) - \arctan (0)}{6}$

Step 8

Evaluate the limit.

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Step 8.1

Move the term $\frac{1}{6}$ outside of the limit because it is constant with respect to $t$ .

$\frac{1}{6} lim_{t \to \infty} \arctan (\frac{2 t}{3}) - \arctan (0)$

Step 8.2

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $\infty$ .

$\frac{1}{6} (lim_{t \to \infty} \arctan (\frac{2 t}{3}) - lim_{t \to \infty} \arctan (0))$

Step 8.3

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{1}{6} (\infty - lim_{t \to \infty} \arctan (0))$

Step 8.4

Substitute $t$ for $\frac{2 t}{3}$ and let $t$ approach $\infty$ since $lim_{t \to \infty} \frac{2 t}{3} = \infty$ .

$\frac{1}{6} (lim_{t \to \infty} \arctan (t) - lim_{t \to \infty} \arctan (0))$

Step 8.5

The limit as $t$ approaches $\infty$ is $\frac{π}{2}$ .

$\frac{1}{6} (\frac{π}{2} - lim_{t \to \infty} \arctan (0))$

Step 8.6

Evaluate the limit of $\arctan (0)$ which is constant as $t$ approaches $\infty$ .

$\frac{1}{6} (\frac{π}{2} - \arctan (0))$

Step 8.7

Simplify the answer.

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Step 8.7.1

Simplify each term.

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Step 8.7.1.1

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

$\frac{1}{6} (\frac{π}{2} - 0)$

Step 8.7.1.2

Multiply $- 1$ by $0$ .

$\frac{1}{6} (\frac{π}{2} + 0)$

Step 8.7.2

Add $\frac{π}{2}$ and $0$ .

$\frac{1}{6} \cdot \frac{π}{2}$

Step 8.7.3

Multiply $\frac{1}{6} \cdot \frac{π}{2}$ .

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Step 8.7.3.1

Multiply $\frac{1}{6}$ by $\frac{π}{2}$ .

$\frac{π}{6 \cdot 2}$

Step 8.7.3.2

Multiply $6$ by $2$ .

$\frac{π}{12}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{12}$

Decimal Form:

$0.26179938 \dots$