Calculus Examples

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

Step 1

Split the integral at $0$ and write as a sum of limits.

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

Step 2

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

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

Step 3

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

$lim_{t \to - \infty} \arctan (x)]_{t}^{0} + lim_{t \to \infty} \int_{0}^{t} \frac{1}{1 + x^{2}} d x$

Step 4

Evaluate $\arctan (x)$ at $0$ and at $t$ .

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

Step 5

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

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

Step 6

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

$lim_{t \to - \infty} \arctan (0) - \arctan (t) + lim_{t \to \infty} \arctan (x)]_{0}^{t}$

Step 7

Evaluate $\arctan (x)$ at $t$ and at $0$ .

$lim_{t \to - \infty} \arctan (0) - \arctan (t) + lim_{t \to \infty} \arctan (t) - \arctan (0)$

Step 8

Evaluate the limits.

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Step 8.1

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

$lim_{t \to - \infty} \arctan (0) - lim_{t \to - \infty} \arctan (t) + lim_{t \to \infty} \arctan (t) - \arctan (0)$

Step 8.2

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

$\arctan (0) - lim_{t \to - \infty} \arctan (t) + lim_{t \to \infty} \arctan (t) - \arctan (0)$

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

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

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

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 $0$ .

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

Step 8.7.1.2

Multiply $\frac{π}{2}$ by $1$ .

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

Step 8.7.1.3

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

$0 + \frac{π}{2} + \frac{π}{2} - 0$

Step 8.7.1.4

Multiply $- 1$ by $0$ .

$0 + \frac{π}{2} + \frac{π}{2} + 0$

Step 8.7.2

Combine the numerators over the common denominator.

$\frac{π + π}{2}$

Step 8.7.3

Add $π$ and $π$ .

$\frac{2 π}{2}$

Step 8.7.4

Cancel the common factor of $2$ .

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Step 8.7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 8.7.4.2

Divide $π$ by $1$ .

$π$

Step 9

The result can be shown in multiple forms.

Exact Form:

$π$

Decimal Form:

$3.14159265 \dots$