Calculus Examples

$\int_{0}^{5} \frac{1}{25 + x^{2}} d x$

Step 1

Rewrite $25$ as $5^{2}$ .

$\int_{0}^{5} \frac{1}{5^{2} + x^{2}} d x$

Step 2

The integral of $\frac{1}{5^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{5} \arctan (\frac{x}{5})]_{0}^{5}$ .

$\frac{1}{5} \arctan (\frac{x}{5})]_{0}^{5}$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{1}{5}$ and $\arctan (\frac{x}{5})$ .

$\frac{\arctan (\frac{x}{5})}{5}]_{0}^{5}$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{\arctan (\frac{x}{5})}{5}$ at $5$ and at $0$ .

$(\frac{\arctan (\frac{5}{5})}{5}) - \frac{\arctan (\frac{0}{5})}{5}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{\arctan (\frac{5}{5})}{5} - \frac{\arctan (\frac{0}{5})}{5}$

Step 3.2.2.1.2

Rewrite the expression.

$\frac{\arctan (1)}{5} - \frac{\arctan (\frac{0}{5})}{5}$

Step 3.2.2.2

Cancel the common factor of $0$ and $5$ .

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Step 3.2.2.2.1

Factor $5$ out of $0$ .

$\frac{\arctan (1)}{5} - \frac{\arctan (\frac{5 (0)}{5})}{5}$

Step 3.2.2.2.2

Cancel the common factors.

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Step 3.2.2.2.2.1

Factor $5$ out of $5$ .

$\frac{\arctan (1)}{5} - \frac{\arctan (\frac{5 \cdot 0}{5 \cdot 1})}{5}$

Step 3.2.2.2.2.2

Cancel the common factor.

$\frac{\arctan (1)}{5} - \frac{\arctan (\frac{5 \cdot 0}{5 \cdot 1})}{5}$

Step 3.2.2.2.2.3

Rewrite the expression.

$\frac{\arctan (1)}{5} - \frac{\arctan (\frac{0}{1})}{5}$

Step 3.2.2.2.2.4

Divide $0$ by $1$ .

$\frac{\arctan (1)}{5} - \frac{\arctan (0)}{5}$

Step 3.3

Simplify.

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Step 3.3.1

Combine the numerators over the common denominator.

$\frac{\arctan (1) - \arctan (0)}{5}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

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

$\frac{\frac{π}{4} - \arctan (0)}{5}$

Step 3.3.2.2

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

$\frac{\frac{π}{4} - 0}{5}$

Step 3.3.2.3

Multiply $- 1$ by $0$ .

$\frac{\frac{π}{4} + 0}{5}$

Step 3.3.3

Add $\frac{π}{4}$ and $0$ .

$\frac{\frac{π}{4}}{5}$

Step 3.3.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{π}{4} \cdot \frac{1}{5}$

Step 3.3.5

Multiply $\frac{π}{4} \cdot \frac{1}{5}$ .

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Step 3.3.5.1

Multiply $\frac{π}{4}$ by $\frac{1}{5}$ .

$\frac{π}{4 \cdot 5}$

Step 3.3.5.2

Multiply $4$ by $5$ .

$\frac{π}{20}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{20}$

Decimal Form:

$0.15707963 \dots$

Step 5