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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.

1)

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)

2) ...

Solution Summary

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.

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... A) Trigonometric substitution. B) Using previously developed solutions. C) Integration by parts. ... C) The function being integrated converges. ...

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... the product of Sin and Cos terms in the above we can make use of the trig identities below. ... Proceeding to solve this (ie carrying out the integration part). ...