APM - Q3 - Trig. integrals
Flashcards
2022-03-04
What is
\[\int \frac{\cos x - \sin x}{\cos x + \sin x}\]
?
\[\ln(\cos x + \sin x) + c\]
Because of the chain rule.
If you have
\[I _ 1 = \int\frac{\cos x}{a\cos x + b \sin x} \text{d}x\]
\[I _ 2 = \int\frac{\sin x}{a\cos x + b\sin x}\]
then what could you do to make a nice integral by combining them that you could evaluate?
\[aI_1 + bI_2 = \int \frac{a\cos x + b\sin x}{a\cos x + b \sin x} \text{d}x\]
\[bI_1 - aI_2 = \int \frac{b \cos x - a \sin x}{a \cos x + b \sin x} \text{d}x\]