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Note: I have used int to denote the integral sign.
I have used several trig identities.
I have assumed that the student is familiar with the integration of basic trig functions such as sinx , cos x , secx *tan x, sec2 x, cosec x and so on. If the student is not familiar with these, it is recommended that he/she learn them.
Let sinx = u, du = cos x dx
I = int (sin^6 x cos ^3 x) dx = int (sin^6 x (1 - sin ^2 x) cos xdx = int u^6 (1 - u^2) du = int (u^6 - u^8) du = u^7/7 - u^9/9 = 1/7* sin^7 (x) - 1/9 * sin^9 (x)
= 1/7* sin^7 (x) - 1/9 * sin^9 (x)
This solution contains the integration of several complicated trigonometric functions. Please download this set, if you need to improve your integration skills using various techniques. I have provided step-by-step solutions to each of the question.