Show :
`int_(a)^(b)f(kx)dx=(1)/(k)int_(ka)^(kb)f(x)dx`

If f(x) and g(x) be continuous in the interval [0, a] and satisfy the conditions f(x)=f(a-x) and g(x)+g(a-x)=a , then show that int_(0)^(a)f(x)g(x)dx=(a)/(2)int_(0)^(a)f(x)dx . Hence find the value of int_(0)^(pi)xsinxdx .

If int_(a)^(b)f(x)dx=int_(a)^(b)phi(x)dx , then-

Prove that int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx.

Prove that, int_(a)^(b)f(a+b-x)dx=int_(a)^(b)f(x)dx .

Let the definite integral be defined by the formula int_(a)^(b)f(x)dx=(b-a)/2(f(a)+f(b)) . For more accurate result, for c epsilon (a,b), we can use int_(a)^(b)f(x)dx=int_(a)^(c)f(x)dx+int_(c)^(b)f(x)dx=F(c) so that for c=(a+b)/2 we get int_(a)^(b)f(x)dx=(b-a)/4(f(a)+f(b)+2f(c)) . If f''(x)lt0 AA x epsilon (a,b) and c is a point such that altcltb , and (c,f(c)) is the point lying on the curve for which F(c) is maximum then f'(c) is equal to

Statement-I : The value of the integral int_((pi)/(6))^((pi)/(3))(dx)/(1+sqrt(tanx)) is equal to (pi)/(6) . Statement-II : int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx

Show that int_(0)^(a)f(x)g(x)dx=2int_(0)^(a)f(x)dx , if f and g are defined as f (x) = f (a - x) and g(x) + g(a - x) = 4

Show that, int_(0)^(pi)xf(sinx)dx=(pi)/(2)int_(0)^(pi)f(sinx)dx .

If f(x)is integrable function in the interval [-a,a] then show that int_(-a)^(a)f(x)dx=int_(0)^(a)[f(x)+f(-x)]dx.

The number of positive continuous f(x) defined in [0,1] for with I_(1)=int_(0)^(1)f(x)dx=1,I_(2)=int_(0)^(1)xf(x)dx=a , I_(3)=int_(0)^(1)x^(2)f(x)dx=a^(2) is /are