[int_(-a)^(a)f(x)dx=2int_(0)^(a)f(x),," if "f" is even function "],[=0,," if "f" is an odd function "]

int_(-a)^(a)f(x)dx= 2int_(0)^(a)f(x)dx, if f is an even function 0, if f is an odd function.

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)^(a)f(x)dx =2 int_(a)^(0) f(x)dx, if f(x) is even funtion =0 , if f(x) is off fuction.

Prove that underset(-a)overset(a)int f(x)dx={{:(,2underset(0)overset(a)int f(x)dx,"if f(x) is an even function"),(,0,"if f(x) is an odd function"):} and hence evaluate underset(-pi//2)overset(pi//2)int (x^(3)+x cos x)dx.

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 (c) int_(0)^(pi)|cosx|dx .

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 .

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 (a) int_(-1)^(1) sin^(5)x cos^(4)xdx .

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 (d) int_(-pi//2)^(pi//2)tan^(9) xdx .