Question

What is the antiderivative of `F(x) = xcosx`?

Answer 1

`intxcosxdx = xsinx+cosx+C`

Explanation:

For this problem, we will use the integration by parts formula

`intudv = uv-intvdu`

along with the following:

• `d/dxx = 1`
• `d/dxsinx = cosx`
• `intsinxdx = -cosx+C`

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Let `u = x` and `dv = cosxdx`
Then `du = dx` and `v = sinx`

Applying the formula, we have

`intxcosxdx = xsinx - intsinxdx`

`=xsinx - (-cosx)+C`

`=xsinx+cosx+C`

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Checking our answer, we find that `d/dx(xsinx+cosx+C)=xcosx`