# Unusual Constant appearing for definite integral

I’m learning integration and have come across a question in a textbook. I arrived to the same answer as the textbook itself, but I know it is not correct. It starts as part a) show that $$\frac{d}{dx} tan^{-1} \left(\frac{3tan(x)}{2} \right)= \frac{6}{5sin^2(x)+4}$$ Hence, find area bounded by curve from $$\dfrac{1}{5sin^2(x) + 4}$$ from $$x=0$$ to $$x=7$$. Clearly using part A to integrate and substituting 7 and 0, I arrive to 0.153 square units, in line with the textbook. Yet, by using desmos to graph the curve, using lower rectangles, minimum area has to be $$0.111111 \times 7 = 0.77777$$ units squared. I put the integral on integral calculator and it yielded $$0.153 + \dfrac{\pi}{3}$$, without explaining where this constant came from. I am unfamiliar with constants in definite integrals, and struggled to find anything online to answer this. Thanks.

(Note: I am not asking for homework help, Im just asking about how to find constants in these definite integrals, particularly in a question like this, which the textbook itself overlooked. Thank you. Any help is appreciated.)

• You have been around for more than a year. Haven't you yet noticed that you are supposed to use MathJax around here? Commented Mar 27, 2021 at 10:42

This happens because $$\tan x$$ is not defined at the points $$x = \frac{\pi}{2}, \frac{3\pi}{2}$$ which lie in $$(0,7)$$ so $$\tan^{-1} \left(\frac{3\tan(x)}{2} \right)$$ becomes discontinuous at these points.
This is why you have to split the interval into sub-intervals $$(0,\frac{\pi}{2}), (\frac{\pi}{2}, \frac{3\pi}{2}), (\frac{3\pi}{2},7)$$ while substituting the limits. $$\Delta = \dfrac16 \left[ \tan^{-1} \left(\frac{3\tan(x)}{2} \right)_0^\frac{\pi^-}{2} + \tan^{-1} \left(\frac{3\tan(x)}{2} \right)_\frac{\pi^+}{2}^\frac{3\pi^-}{2} + \tan^{-1} \left(\frac{3\tan(x)}{2} \right)_\frac{3\pi^+}{2}^7 \right]$$
$$\Delta = \dfrac16 \left[ \dfrac{\pi}{2} - 0+ \dfrac{\pi}{2} - \left(- \dfrac{\pi}{2} \right) + \tan^{-1}\left(\frac{3\tan7}{2} \right) - \left(- \dfrac{\pi}{2} \right) \right]$$
$$\Delta = \dfrac{\pi}{3}+\dfrac16 \tan^{-1}\left(\frac{3\tan7}{2} \right)$$