00:01
Hi, so we are given from the question that to evaluate the closed integral of with respect to r, x cosine of y da with respect to y is equals to x, y is equals to 0 and x is equals to pi.
00:24
So, first of all we need to break this integral in such a way that the double integration r, x cosine of y da this should be equals to, so as you can see y is equals to x.
00:40
So, once y is equals to 0, x will also be equals to 0 and already x is nothing but equals to pi and here y is equals to x.
00:50
So, two limits have been got here while this area will be in respect to dx and dy substituting above, our required integral will become integration 0 to pi for x, integration 0 to x for y as y is equals to x and here it should be the x cosine of y dx dy.
01:15
So, first integrate with respect to y, we get integration 0 to pi, x will be as it is, integration of cosine of y will become sine of y and the limit will go from 0 to x dx.
01:34
So, this will be goes to integration 0 to pi x, let's plug in the values, so sine of x minus sine of 0 dx.
01:46
So, sine 0 is nothing but 0 and hence you will get integration 0 to pi x into the sine of x dx.
01:56
Now, to solve this, we have to use integration by parts formula that is integration v into u, this can be equals to u common integration of v dx minus integral of into the bracket derivative of the first term that is u times of integration of v dx out of the bracket dx.
02:14
So, how we select u and v by using euler's rule where i stand for inverse logarithmic arithmetic trigonometry exponential.
02:27
So, clearly x is arithmetic here and sine of x is a trigonometry, so arithmetic come first as the trigonometry that means this x will be considered u and sine will be considered as v.
02:40
So, according to formulae, this should become equals to u common that means x common integration of sine of x dx will be there minus integration of into the bracket d by dx of u that means x dot integration of sine of x dx out of the bracket dx.
03:02
So, this will be goes to x integration of sine will be minus of cosine of x minus of integration of into the bracket derivative of x will going to be 1 into integration of sine x will be minus cosine of x and out of the bracket dx minus minus will get plus here...