Manipal MET2012MathematicsDefinite Integration
If I(m, n)= ₀^1 t^m(1+t)^n d t , then the expression for I(m, n) in terms of I(m+1, n-1) is
Options
- A2^n m+1 - n m+1 I(m+1, n-1)
- Bn m+1 I(m+1, n-1)
- C2^n m+1 + n m+1 I(m+1, n-1)
- Dm m+1 I(m+1, n-1)
Correct answer
A. 2^n m+1 - n m+1 I(m+1, n-1)
Step-by-step solution
Here, I(m, n)= ₀^1 t^m(1+t)^n d t [We apply integration by parts, taking (1+t)^n as first and t^m as second function] aligned & I(m, n)= [(1+t)^n t^ m+1 m+1 ]₀^1 & - ₀^1 n(1+t)^ n-1 t^ m+1 m+1 d t aligned aligned & = 2^n m+1 - n m+1 ₀^1(1+t)^ n-1 t^ m+1 d t & I(m, n)= 2^n m+1 - n m+1 I(m+1, n-1) aligned