" Find "int_(0)^(a)y^(2)dx" whese "x=a" (as "a,y=b" sine "

int_(0)^(1)y dx where x=cos2y

int_(0)^(a)y^(2)dx where x=acosthetaandy=bsintheta

int_(0)^((pi)/(2))y^(2)dx where y = sin x

int_(0)^(1)x(1-x)^(9y)dx

int_(a)^(2a)xy dx where x^(2)-y^(2)=a^(2) and y>=0

Solve the differential equation (dy)/(dx) = y + int_(0)^(1)y(x)dx , given that the value of y is 1, when x = 0.

Solve the differential equation (dy)/(dx) = y + int_(0)^(1)y(x)dx , given that the value of y is 1 , when x = 0 .

Prove that int_(0)^(a) f(x) dx = int_(0)^(a) f(a - x)dx and hence evaluate the following: (f) int_(0)^(pi)(xdx)/(a^(2)cos^(2)x + b^(2)sin^(2)x)

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 .