Question

What is `f(x) = int xsinx dx` if `f(pi/3) = -2`?

Answer 1

`f(x) = -xcosx + sinx - 2.34`, approximately.

Explanation:

Use integration by parts.

Let `u = x` and `dv = sinxdx`.

So, `du = dx` and `v = -cosx`

Now, use the integration by parts formula `int(udv) = uv - int(vdu)`.

`int(xsinx) = -xcosx - int(-cosx)`

`f(x) = -xcosx + sinx + C`

Now, all we have to do is determine the value of `C`.

We know that when `x = pi/3, y = -2`, so:

`-2 = -pi/3cos(pi/3) + sin(pi/3) + C`

`-2 + pi/3cos(pi/3) + sin(pi/3) = C`

Evaluating using a calculator, you should get `C ~=-2.34`

Hopefully this helps!