 Hi and welcome to the session. I am Pooja and I will help you with the following question. Integrate the function sin raised to the power 8x minus cos raised to the power 8x upon 1 minus 2 sin square x into cos square x. Now we pick in with the solution. We denote this function by i. So we have i is equal to integral sin raised to the power 8x minus cos raised to the power 8x upon 1 minus 2 sin square x into cos square x dx. This is equal to integral. Now we can write sin raised to the power 8x as sin raised to the power 4x to the whole power 2 minus we can write cos raised to the power 8x as cos raised to the power 4x to the whole power 2 upon 1 minus 2 sin square x cos square x dx. This is equal to integral. Now in numerator we can apply the formula for a square minus b square which is equal to a plus b into a minus b. So we get sin raised to the power 4x plus cos raised to the power 4x into sin raised to the power 4x minus cos raised to the power 4x upon 1 minus 2 sin square x cos square x dx since a square minus b square is equal to a plus b into a minus b. Now this is equal to integral. Now we can write sin raised to the power 4x as sin square x to the whole power 2 plus we can write cos raised to the power 4x as cos square x to the whole power 2 into sin square x to the whole power 2 minus cos square x to the whole power 2 upon 1 minus 2 sin square x into cos square x dx. This is equal to integral sin square x to the whole power 2 plus cos square x to the whole power 2. Now adding and subtracting 2 sin square x cos square x we get plus 2 sin square x cos square x minus 2 sin square x cos square x into. Now again we will apply the formula for a square minus b square here. So we will get sin square x minus cos square x into sin square x plus cos square x because we know that the formula for a square minus b square is a minus b into a minus b upon 1 minus 2 sin square x cos square x dx. This is further equal to integral. Now we will apply the formula for a square plus b square plus 2ab here. So a square plus 2ab is equal to a plus b whole square. So we will get sin square x plus cos square x whole square minus 2 sin square x cos square x into sin square x minus cos square x into. Now sin square x plus cos square x is 1. So we will get into 1 upon 1 minus 2 sin square x cos square x dx. We will write here since a plus b whole square is equal to a square plus b square plus 2ab. This is equal to integral. Now sin square x plus cos square x is equal to 1. So we have 1 square minus 2 sin square x cos square x into sin square x minus cos square x upon 1 minus 2 sin square x cos square x dx. Now 1 square is equal to 1. So we can cancel out the common factor in numerator and denominator. So we are left with this is equal to integral sin square x minus cos square x dx. And we can write this as this is equal to integral sin square x dx minus integral cos square x dx. This is equal to integral. Now we know that sin square x is equal to 1 minus cos square x. So we get 1 minus cos square x dx minus integral cos square x dx. Since sin square x is equal to 1 minus cos square x. This is further equal to integral dx minus integral cos square x dx minus integral cos square x dx. And this is equal to integral dx minus 2 integral cos square x dx. This is equal to integral dx minus 2 integral. Now we know that cos square x is equal to 1 minus cos 2x upon 2 dx. Since cos square x is equal to 1 minus cos 2x upon 2. Canceling out 2 denominator and numerator we get this is equal to integral dx minus integral dx minus into minus becomes plus integral cos 2x dx. Now again cancelling out these both common terms we get this is equal to integral cos 2x dx. This is equal to now integrating cos 2x we get minus sin 2x upon 2 plus c. So we get our answer as minus 1 by 2 sin 2x plus c. Hope you have understood the solution. Take care and bye.