
River mechanics problem
I'm having trouble with this question and I appreciate anyone's help.
A man swims across a straight river of uniform width W, starting from a point O on one bank of the river. The velocity of the river at a distance y from the bank is
u(y)=ay(W − y), where a is a positive constant. The man travels at a constant speed v relative to the current and steers a course set at a
constant angle θ (0 < θ < pi) to the downstream direction.
(a) Show that the velocity of the man relative to a Cartesian coordinate system with origin at O, i pointing in the downstream direction, and j pointing across the river is given by (u + v cos θ)i + (v sin θ)j. I've done this part.
(b) At what time does the man reach the other bank?
I get t = W/v.sinθ  I think that's correct.
(c) Show that when the man has reached the other bank, the downstream distance he has travelled is equal to aW^3/6v.sinθ + W.cot θ.
This is the part i'm having problems with. I thought about replacing t from part (b) into the integrated i component from the velocity in part (a). I'm not sure what to do with the u(y) though.

When the swimmer has swum for time t, he is distance $\displaystyle (v\sin{\theta})t$ away from the shore. The downstream current speed at that spot is $\displaystyle a(v\sin{\theta})t[W(v\sin{\theta})t]=aW(v\sin{\theta})ta(v^2\sin^2{\theta})t^2$. So the total horizontal component of the man’s velocity is $\displaystyle aW(v\sin{\theta})ta(v^2\sin^2{\theta})t^2+v\cos{\theta}$.
Hence, the total horizontal displacement of the man is
$\displaystyle \color{white}.\quad.$ $\displaystyle \int_0^{\frac{W}{v\sin{\theta}}}{[aW(v\sin{\theta})ta(v^2\sin^2{\theta})t^2+v\cos{\theta}]}\,\mathrm{d}t$