Find an antiderivative of each of the following:sin(2x + π4)a cos(πx)b sin( 2πx3)c

Sure, let's find the antiderivatives of each function:

a) ∫sin(2x + π/4) dx

We can use the formula ∫sin(ax+b) dx = -1/a cos(ax+b) + C. Here, a = 2 and b = π/4. So, the antiderivative is:

-1/2 cos(2x + π/4) + C

b) ∫cos(πx) dx

We can use the formula ∫cos(ax) dx = 1/a sin(ax) + C. Here, a = π. So, the antiderivative is:

1/π sin(πx) + C

c) ∫sin(2πx^3) dx

This one is a bit more complicated because it involves a composite function. We can't directly integrate it using elementary functions. However, we can express it in terms of a definite integral:

∫sin(2πx^3) dx = ∫sin(u) du/(6x^2) from 0 to 2πx^3

This integral doesn't have a standard elementary antiderivative. It can be expressed in terms of special functions like the sine integral Si(x), but that's beyond the scope of basic calculus.