Calculus Examples

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

Step 1

Write the integral as a limit as $t$ approaches $\infty$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Evaluate the limit.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

$\frac{π}{2} - \arctan (0)$

Step 5.4

Simplify the answer.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

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

$\frac{π}{2} - 0$

Step 5.4.1.2

Multiply $- 1$ by $0$ .

$\frac{π}{2} + 0$

Step 5.4.2

Add $\frac{π}{2}$ and $0$ .

$\frac{π}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{2}$

Decimal Form:

$1.57079632 \dots$