Calculus Examples

$\int_{0}^{2} \frac{1}{1 + (\frac{x^{2}}{4})} d x$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Simplify.

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Write $1$ as a fraction with a common denominator.

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

Step 2.1.2

Combine the numerators over the common denominator.

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

Step 2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3

Multiply $\frac{4}{4 + x^{2}}$ by $1$ .

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

Step 3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 4

Rewrite $4$ as $2^{2}$ .

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

Step 5

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

$4 (\frac{1}{2} \arctan (\frac{x}{2})]_{0}^{2})$

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{1}{2}$ and $\arctan (\frac{x}{2})$ .

$4 (\frac{\arctan (\frac{x}{2})}{2}]_{0}^{2})$

Step 6.2

Substitute and simplify.

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Step 6.2.1

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

$4 ((\frac{\arctan (\frac{2}{2})}{2}) - \frac{\arctan (\frac{0}{2})}{2})$

Step 6.2.2

Simplify.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$4 (\frac{\arctan (\frac{2}{2})}{2} - \frac{\arctan (\frac{0}{2})}{2})$

Step 6.2.2.1.2

Rewrite the expression.

$4 (\frac{\arctan (1)}{2} - \frac{\arctan (\frac{0}{2})}{2})$

Step 6.2.2.2

Cancel the common factor of $0$ and $2$ .

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Step 6.2.2.2.1

Factor $2$ out of $0$ .

$4 (\frac{\arctan (1)}{2} - \frac{\arctan (\frac{2 (0)}{2})}{2})$

Step 6.2.2.2.2

Cancel the common factors.

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Step 6.2.2.2.2.1

Factor $2$ out of $2$ .

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

Step 6.2.2.2.2.2

Cancel the common factor.

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

Step 6.2.2.2.2.3

Rewrite the expression.

$4 (\frac{\arctan (1)}{2} - \frac{\arctan (\frac{0}{1})}{2})$

Step 6.2.2.2.2.4

Divide $0$ by $1$ .

$4 (\frac{\arctan (1)}{2} - \frac{\arctan (0)}{2})$

Step 7

Simplify.

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Step 7.1

Combine the numerators over the common denominator.

$4 \frac{\arctan (1) - \arctan (0)}{2}$

Step 7.2

Simplify each term.

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Step 7.2.1

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

$4 \frac{\frac{π}{4} - \arctan (0)}{2}$

Step 7.2.2

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

$4 \frac{\frac{π}{4} - 0}{2}$

Step 7.2.3

Multiply $- 1$ by $0$ .

$4 \frac{\frac{π}{4} + 0}{2}$

Step 7.3

Add $\frac{π}{4}$ and $0$ .

$4 \frac{\frac{π}{4}}{2}$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Factor $2$ out of $4$ .

$2 (2) \frac{\frac{π}{4}}{2}$

Step 7.4.2

Cancel the common factor.

$2 \cdot 2 \frac{\frac{π}{4}}{2}$

Step 7.4.3

Rewrite the expression.

$2 \frac{π}{4}$

Step 7.5

Combine $2$ and $\frac{π}{4}$ .

$\frac{2 π}{4}$

Step 7.6

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

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Step 7.6.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 7.6.2

Cancel the common factors.

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Step 7.6.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 7.6.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 7.6.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{2}$

Decimal Form:

$1.57079632 \dots$

Step 9