<s>[INST] Infinite limits on integration: acceptable notation So you have an integral like
$$

\int_{-\infty}^\infty{ \frac{dx}{1+4 x^2} }

$$
Schaum's Calculus 5e recommends you write this as
$$
\lim_{a \to -\infty} \int_a^b{ \frac{dx}{1+4 x^2} } + \lim_{c \to \infty} \int_b^c{ \frac{dx}{1+4 x^2} }
$$
Where b is chosen as a point where f(x) is defined.
Choosing b=0, you then get
$$
\lim_{a \to -\infty} \frac{1}{4} \int_a^0{ \frac{dx}{\frac{1}{4}+x^2} } + \lim_{c \to \infty} \frac{1}{4} \int_0^c{ \frac{dx}{\frac{1}{4}+x^2} }
$$
$$
\lim_{a \to -\infty} \frac{1}{2} \tan^{-1}{ 2x } |_a^0 + \lim_{c \to \infty} \frac{1}{2} \tan^{-1}{ 2x } |_0^c
$$
$$
0 - \lim_{a \to -\infty} \frac{1}{2} \tan^{-1}{ 2a } + \lim_{c \to \infty} \frac{1}{2} \tan^{-1}{ 2c } - 0
$$
$$
= \frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2}
$$
Would it be correct to shortcut this as


*

*Take the indefinite integral with no limits
$$
\frac{1}{4} \int{ \frac{dx}{ \frac{1}{4} + x^2} } = \frac{1}{2} \tan^{-1}{ 2x }
$$

*Evaluate
$$
= \frac{1}{2} \lim_{a \to -\infty} \lim_{b \to \infty} \tan^{-1}{ 2x } |_a^b
$$
$$
= \frac{1}{2} \lim_{a \to -\infty} \lim_{b \to \infty} \left( \tan^{-1}{ 2b } - \tan^{-1}{ 2a } \right)
$$
$$
= \frac{1}{2} \left( \frac{\pi}{2} + \frac{\pi}{2} \right) = \frac{ \pi }{2}
$$
So my question is surrounding limit notation and evaluation.  Is the above notation ok, or is it completely necessary to break it up into 2 limits?
 [/INST] To see why they have this approach, try 
$$\int_{-\infty}^\infty{ \frac{8x \;dx}{1+4 x^2} } $$ 
using the recommended method [the indefinite integral is $\log(1+4 x^2) + \text{constant}$], and then using your suggestion, and alternatively what would happen if you had to work out
$$\lim_{c \to +\infty} \int_{-c}^c { \frac{8x \;dx}{1+4 x^2} }.$$
You would be looking with the recommended method  at something like $\infty-\infty$, with your suggestion of doing the upper limit first at $\lim_{a \to -\infty} +\infty = +\infty$,  and with the alternative method at $\lim_{c \to +\infty} 0 = 0$.  In fact you want the alarm bells of the difference between two infinities.
</s>