Calculus Examples

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

Step 1

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

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

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

Step 4.2

Substitute and simplify.

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Step 4.2.1

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Rewrite the expression.

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

Step 4.2.2.2

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

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Step 4.2.2.2.1

Factor $2$ out of $0$ .

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

Step 4.2.2.2.2

Cancel the common factors.

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Step 4.2.2.2.2.1

Factor $2$ out of $2$ .

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

Step 4.2.2.2.2.2

Cancel the common factor.

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

Step 4.2.2.2.2.3

Rewrite the expression.

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

Step 4.2.2.2.2.4

Divide $0$ by $1$ .

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

Step 4.3

Simplify.

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Step 4.3.1

Combine the numerators over the common denominator.

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

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

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

$\frac{\frac{π}{4} - \arctan (0)}{2}$

Step 4.3.2.2

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

$\frac{\frac{π}{4} - 0}{2}$

Step 4.3.2.3

Multiply $- 1$ by $0$ .

$\frac{\frac{π}{4} + 0}{2}$

Step 4.3.3

Add $\frac{π}{4}$ and $0$ .

$\frac{\frac{π}{4}}{2}$

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{π}{4} \cdot \frac{1}{2}$

Step 4.3.5

Multiply $\frac{π}{4} \cdot \frac{1}{2}$ .

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Step 4.3.5.1

Multiply $\frac{π}{4}$ by $\frac{1}{2}$ .

$\frac{π}{4 \cdot 2}$

Step 4.3.5.2

Multiply $4$ by $2$ .

$\frac{π}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{8}$

Decimal Form:

$0.39269908 \dots$

Step 6