 Hello and welcome to the session. Let us discuss the following question. Question says, using the properties of integrals, evaluate the integral, definite integral from 0 to pi upon 2 cos x raised to the power 5 upon sin x raised to the power 5 plus cos x raised to the power 5 dx. First upon, let us understand that integral from 0 to a fx dx is equal to integral from 0 to a f a minus x dx. We will use this property of definite integral as our key idea to solve the given question. Let us now start with the solution. We are given integral 0 to pi upon 2 cos x raised to the power 5 upon sin x raised to the power 5 plus cos x raised to the power 5 dx. Now using the property given in key idea, this integral can be written as 0 to pi upon 2 cos pi upon 2 minus x raised to the power 5 upon sin pi upon 2 minus x raised to the power 5 plus cos pi upon 2 minus x raised to the power 5 dx. Now we know cos pi upon 2 minus x is equal to sin x and sin pi upon 2 minus x is equal to cos x. So we can write this integral as sin x raised to the power 5 upon cos x raised to the power 5 plus sin x raised to the power 5 dx. Now let us assume that this given integral is equal to i and i is also equal to definite integral from 0 to pi upon 2 sin x raised to the power 5 upon cos x raised to the power 5 plus sin x raised to the power 5 dx. Now let us name these expressions as 1 and 2. Adding expressions 1 and 2 we get 2i is equal to definite integral from 0 to pi upon 2 cos x raised to the power 5 plus sin x raised to the power 5 upon cos x raised to the power 5 plus sin x raised to the power 5 dx. Now numerator and denominator of this integrant will get cancelled and we get 2i is equal to definite integral from 0 to pi upon 2 dx. Now we know integral of 1 is x. So we get here x and limits are 0 to pi upon 2 and here we can write 2i. Now substituting the limits in the answer we get pi upon 2 minus 0 is equal to 2i. Now dividing both sides by 2 we get i is equal to pi upon 4. We know i is equal to given integral that is definite integral from 0 to pi upon 2 cos x raised to the power 5 upon sin x raised to the power 5 plus cos x raised to the power 5 dx and this integral is equal to pi upon 4. So this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.