Definite Integral..help me out!

In these type of problems , the aim is to part - integrate the integral in such a way that the second function is the one whose " value " , i . e , in this case , the power of " t ^m " is seen to have increased . In the solution below , the limits are from " 0 " to " 1 " .

I ( m , n ) = ∫ t ^m ( 1 + t ) ⁿ dt

= ( 1 + t ) ⁿ ∫ t ^m dt - ∫ d ( 1 + t ) ⁿdt [ ∫ t ^m dt ] dt

= [ t ^{m + 1} ( 1 + t ) ⁿm + 1 ] - nm + 1 ∫ t ^{m + 1} ( 1 + t ) ^{n - 1} dt

= 2 ⁿm + 1 - nm + 1 I ( m + 1 , n - 1 )........................After putting the limits .

Hence , the answer is " ( a ) " .

@Ricky - you used 0, 1 as the limits because the integral is strikingly similar to a beta function, isn't it? :P