There are three ways to go about this, presented in increasing order of rigour:

1. Graphical solution.

First look at the two integrands:

The main thing is to recognise one to be the inverse of the other.

Now, draw a tentative graph of, say, f(x). It will resemble the sine curve.

To get at the graph of g(x) is simple, you have to make the y-axis the x-axis. Since f(0) = g(0) =1 and f(1) = g(1) = 0, you can easily see that the area under the graph is the same for both. 

2. Logical solution:

This depends on two facts:

1. f(x) and g(x) are inverses of each other

2. The domain and range of f(x) (and hence of g(x)) is [0,1]. For every t1 belonging to [0,1] we can find t2 belonging to [0,1] such that f(t1) = g(t2).

From this you can easily deduce that the two integrals have the same value

3. No-nonsense solution: