Reduction Formulae and Wallis' Formula

JEE Main Mathematics · Definite Integration · 4 questions, latest first

• The value of k ∈ N for which the integral I_n=∫_0^1(1-x^k)^n d x, n ∈ N, satisfies 147 I_20=148 I_21 is20248 April, Shift 1 · Q11
• Let r_k=(∫_0^1(1-x^7)^k d x)/(∫_0^1(1-x^7)^k+1 d x), k ∈ N. Then the value of Σ_k=1^10 1/(7(r_k-1)) is equal to20246 April, Shift 1 · Q25
• The value of (e^-(π)/4+∫_0^((π)/4) e^-x tan^50 x d x)/(∫_0^((π)/4) e^-x( tan^49 x+ tan^51 x) d x) is202313 April, Shift 2 · Q10
• For m, n>0, let α(m, n)=∫_0^2 t^m(1+3 t)^n d t. If 11 α(10,6)+18 α(11,5)=p(14)^6, then p is equal to ____.202311 April, Shift 1 · Q26