Solve the following in the shortest possible way

a hint !!

this problem  WOULD NOT REQUIRE ANY TYPE OF SUBSTITUTION OR BY PARTS ETC .

THE ULTIMATE INTEGRATION IS JUST A FORMULA  AVAILABLE IN THE FIRST PAGE OF ANY INTEGRATION BOOK .

IT JUST REQUIRES  A LI'L BIT OF IMAGINATION !!

here a>b>0



let a side X forms triangle wid a and b den







dis gives



a^2+b^2-2abcosx=X^2



ryt



now



(sinx/X)^2=k(constant)................constant=(1/4R^2)



thus given integral is integral of a constant



integral=pi x constant



thats it !!!!!!1



rest ........still m solving

You are very close !!

Please complete the remaining steps !!

Now I am providing the complete solution to this problem :

For ease of understansing , let us change the variable x to C

Now consider a triangle whose sides are a, b ( fixed )(a>b) and their intermediate angle C is variable .

Now it is clear that the Dr. of the given integrand =

so the integral  now becomes

Now applying Sine rule , we know that

again we have that       dC = - dA -dB

Now let us imagine what would be the limits of A & B as C varies from 0 to ( with a>b ) .

when C->0  , A-> & B->0

again when C->, A->0 & B->0 { Draw a figure with a>b & observe the above result yourself !! }

so the integral becomes now

The last integral is clearly zero !!

So the ultimate integral is

 but a is constant ; so the result is

=

( hence proved )