1) Integrate the following functions:f(x)= (sinx+cosx)^2

To integrate the function f(x) = (sinx + cosx)^2, we can first expand the square:

f(x) = sin^2(x) + 2sin(x)cos(x) + cos^2(x)

We know that sin^2(x) + cos^2(x) = 1, and 2sin(x)cos(x) is the double angle formula for sin(2x). So we can rewrite the function as:

f(x) = 1 + sin(2x)

Now we can integrate term by term:

∫f(x) dx = ∫1 dx + ∫sin(2x) dx

The integral of 1 dx is x, and the integral of sin(2x) dx is -1/2 cos(2x). So the integral of f(x) is:

∫f(x) dx = x - 1/2 cos(2x) + C

where C is the constant of integration.