 Hello and welcome to the session. I am Deepika and I am going to help you to solve the following question. The question says, using properties of definite integrals, prove the following. Integrant from 0 to pi x10x over secant x into cosecant x dx is equal to pi square upon 4. So let's start the solution. Let i is equal to given integral that is integral from 0 to pi x10x over secant x into cosecant x dx. Now we know that 10x is equal to sine x over cosecant x is equal to 1 over sine x. So we have i is equal to integral from 0 to pi x sine x into sine x over cosecant x dx. Again, we know that secant x is equal to 1 over cosecant x. So we have i is equal to integral from 0 to pi x sine square x dx. Now we have one property of definite integral which is integral from 0 to a f of x dx is equal to integral from 0 to a f of a minus x dx. So by using this property, we have integral from 0 to pi x sine square x dx is equal to integral from 0 to pi pi minus x into sine square pi minus x dx. And this is equal to integral from 0 to pi pi minus x into sine square x dx. And this is again equal to integral from 0 to pi pi sine square x dx minus integral from 0 to pi x sine square x dx. Therefore, we have i is equal to integral from 0 to pi pi sine square x dx minus i or 2i is equal to integral from 0 to pi pi sine square x dx. Now we know that sine square x is equal to 1 minus cosecant x over 2. So we have 2i is equal to pi over 2 integral from 0 to pi 1 minus cosecant x dx. And this is equal to pi over 2 into integral of 1 minus cosecant x dx and integral of 1 minus cosecant x dx is equal to x minus sine 2x over 2. Now the limits are from 0 to pi and this is equal to pi over 2 into pi minus sine 2 pi over 2 minus 0 minus sine 0 over 2. And this is again equal to pi over 2 into pi. So we have 2i is equal to pi square over 2. Therefore, i is equal to pi square over 4. Hence, we have proved that the given integral is equal to pi square over 4. So this completes our session. I hope the solution is clear to you. Bye and have a nice day.