00:01
In this problem, we have to evaluate the definite integral 0 up to pi over 2, cause of x times second of pi sine of x over 4, d of x using method of substitution.
00:34
Now, let us consider u is equal to pi sine of x divided by 4.
00:46
Now, taking differential on both sides, we get differential of u is equal to pi over 4 times differential of sine of x is, cause of x d of x x now multiplying both sides by four over pi we get four over pi times d u is equal to cause of x d of x now we are going to change the limits of integration that is when x is equal to 0 then u is equal to since sine of 0 is 0 so pi over 4 times sign of 0 is 0 and when x is equal to pi over 2 then u is equal to pi over 2 then u is equal to pi over 4 times sine up pi by 2 2 is equal to 1, so u is equal to pi over 4.
02:15
Now using these substitutions in the given definite integral, we have definite integral, the new limits of integration are from 0 up to pi by 4, second of pi times sine x over 4 is equal to u, and cause of x, d of x, is equal to 4 by pi times du, 4 over pi times du.
03:07
Now using properties of definite integral, we can take pi over 4 outside the integral sign...