Question

Can an integral have two answers?

Answer 1

It is possible only in case of Indefinite Integration.

You can see a single function has two different results. Thus it is possible to have more than one result in Indefinite Integral.

For example,

$\int \frac{1}{(x^2+a^2)}dx=\left(\frac{1}{a}\right){\tan }^{-1}\left(\frac{x}{a}\right)+C=-\left(\frac{1}{a}\right)co{t}^{-1}\left(\frac{x}{a}\right)+C$

Hence, an integral can have two answers.