Improper integrals Calculator

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Example

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Solved example of improper integrals

$\int_0^{\infty}\left(\frac{1}{1+x^2}\right)dx$

Replace the integral's limit by a finite value

$\lim_{c\to\infty }\:\int_{0}^{c}\frac{1}{1+x^2}dx$

Solve the integral applying the formula $\displaystyle\int\frac{x'}{x^2+a^2}dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)$

$\lim_{c\to\infty }\left(\left[\arctan\left(x\right)\right]_{0}^{c}\right)$

Evaluate the definite integral

$\lim_{c\to\infty }\left(\arctan\left(c\right)-\arctan\left(0\right)\right)$

Calculating the arctangent of $0$

$\lim_{c\to\infty }\left(\arctan\left(c\right)+\left(-1\right)\left(0\right)\right)$

Any expression multiplied by $0$ is equal to $0$

$\lim_{c\to\infty }\left(\arctan\left(c\right)\right)$

Simplifying

$\lim_{c\to\infty }\left(\arctan\left(c\right)\right)$

Apply the limit $\lim_{x\to\infty}\arctan(x)=\frac{\pi}{2}$

$\frac{\pi}{2}$

Answer