Prove that `int_(a)^(b) f(x)dx=int_(a)^(c)f(x)dx+int_(c)^(b)f(x)dx`

Prove that int_(a)^(b)(x)dx = int_(a)^(b)f(a+b-x)dx and int_(pi/4)^(pi/3)(dx)/(1+sqrt(tanx)) .

Prove that int_(a)^(b) f(x)dx= int_(a)^(b) f (a+b-x)dx" hence evaluate " int_(0)^(pi/4) log(1+tan x)dx .

Prove that: int_(-a)^(a) f(x) dx = {{:(2int_(0)^(a)f(x)dx, f(x) " is even "),(0, f(x) " is odd"):} and hence Evaluate int_(-t)^(t) sin^(5)(x)cos^(4)(x) dx

Prove that int_(-a)^(a)f(x)dx= {(2int_(0)^(a)f(x)dx ,"if f(x) is even function"),(0, "if f(x) is odd function"):} and hence evaluate int_(-(pi)/(2))^((pi)/(2))sin^(7)xdx

Prove that int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx and hence evaluate int_((pi)/(6))^((pi)/(3))(1)/(1+sqrt(tanx))dx.

Prove that int_(0)^(2a) f(x) dx = 2int_(0)^(a) f(x) dx when f(2a -x) = f(x) and hence evaluate int_(0)^(pi) |cos x| dx .

(a) Prove that int_(0)^(2x) f(x) dx = 2int_(0)^(2x) f(x) dx when f(2a-x) =f(x) and hence evaluate int_(0)^(pi) |cos x| dx . (b) Prove that |{:(-a^(2),ab,ac),(bc,-b^(2),bc),(ca,cb,-c^(2)):}|=4a^(2)b^(2)c^(2) .

a) Prove that int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx" and evaluate "int_(pi//6)^(pi//3)(dx)/(1+sqrt(tanx)) b) Prove that |{:(1+a^(2)-b^(2), 2ab, -2b), (2ab, 1-a^(2)+b^(2), 2a), (2, -2a, 1-a^(2)-b^(2)):}|=(1+a^(2)+b^(2))^(3)

Prove that int_(-a)^(a) dx = {(2int_(0)^(a) f(x) dx, if f(x) "is even"),(0, if f(x) "is odd"):} and hence evaluate (b) int_(-pi//2)^(pi//2) sin^(7) x dx .