Show that `int_0^(npi+v)|sinx|dx=2n+1-cosv ,` where `n` is a positive integer and ,`0<=vltpi`

The value of the integral int_(0)^(pi//2) sin 2n x cot x dx, where n is a positive integer, is

Show that 9^(n+1)-8n-9 is divisible by 64, where n is a positive integer.

Show that 9^(n+1)-8n-9 is divisible by 64, where n is a positive integer.

Statement-1: int_(0)^(npi+v)|sin x|dx=2n+1-cos v where n in N and 0 le v lt pi . Stetement-2: If f(x) is a periodic function with period T, then (i) int_(0)^(nT) f(x)dx=n int_(0)^(T) f(x)dx , where n in N and (ii) int_(nT)^(nT+a) f(x)dx=int_(0)^(a) f(x) dx , where n in N

Show that 7^(n) +5 is divisible by 6, where n is a positive integer

If U_(n)=int_(0)^(pi)(1-cosnx)/(1-cosx)dx where n is positive integer of zero, then The value of U_(n) is a. pi//2 b. pi c. npi//2 d. npi

Evaluate int_(0)^(1)(tx+1-x)^(n)dx , where n is a positive integer and t is a parameter independent of x . Hence , show that ∫ 0 1 ​ x k (1−x) n−k dx= [ n C k ​ (n+1)] P ​ fork=0,1,......n, then P=

Prove that (1+i)^n+(1-i)^n=2^((n+2)/2).cos((npi)/4) , where n is a positive integer.

Prove that (1+i)^n+(1-i)^n=2^((n+2)/2).cos((npi)/4) , where n is a positive integer.

If A= [(3 , -4), (1 , -1) ] , then prove that A^n=[(1+2n , -4n), (n , 1-2n) ] , where n is any positive integer.