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Trigonometry and Integration question?
1) What is the maximum value of y = sin^ 8 x + cos ^14 x . Here I am not looking at a solution using calculus ( ie using dy/dx = 0) I understand a pure trigonometry solution exists. Can someone help
2) ∫[ (cos x) / (cos 4x)] dx
I understand that there is a simpler method to solve than expressing cos 4x in terms of sinx and substituting sinx = t and integrating ?
I would also like to know the minimum value of sin^ 8 x + cos ^14 x
2 Answers
- gianlinoLv 79 years agoFavorite Answer
The maximum is 1. Indeed sin^ 8 x + cos ^14 x <= sin^2 x + cos ^2 x = 1 and that value is attained at multiples of pi/2.
For the minimum I guess you have to differentiate:
dy / dx = sin x cos x * [ 8 sin ^6 x - 14 cos ^ 12 x]
So the minimum is for | sin x / cos^2 x | = (7/4]^[1/6].
For your integral it is indeed very simple!!!
- ?Lv 44 years ago
this may be a substitution question. enable u = sinx because it differentiates into du = cosx dx indispensable[sin²x*cosx dx] = indispensable[u² du] = (u³/3) + c <= sub in u lower back for sinx = (sin³x)/3 + c wish this facilitates.