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?