Calculus Examples

$\int_{- 1}^{1} \frac{1}{1 + y^{2}} d y$

Step 1

Rewrite $1$ as $1^{2}$ .

$\int_{- 1}^{1} \frac{1}{1^{2} + y^{2}} d y$

Step 2

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

$\arctan (y)]_{- 1}^{1}$

Step 3

Evaluate $\arctan (y)$ at $1$ and at $- 1$ .

$\arctan (1) - \arctan (- 1)$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

$\frac{π}{4} - \arctan (- 1)$

Step 4.1.2

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

$\frac{π}{4} - - \frac{π}{4}$

Step 4.1.3

Multiply $- - \frac{π}{4}$ .

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{π}{4} + 1 \frac{π}{4}$

Step 4.1.3.2

Multiply $\frac{π}{4}$ by $1$ .

$\frac{π}{4} + \frac{π}{4}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{π + π}{4}$

Step 4.3

Add $π$ and $π$ .

$\frac{2 π}{4}$

Step 4.4

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

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Step 4.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 4.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 4.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{2}$

Decimal Form:

$1.57079632 \dots$

Step 6