Question

If θ f(x) = `int_0^x t sin t  dt` then `f^1(x)` is

Options

• `cosx + x sinx`

• `x sin x`

• `x cos x`

• `sin x + x cos x`

Answer 1

`x sin x`

Explanation:

R.H.S = `f(x) = int_0^x t sin t  dt`

Integrating, by parts taking 't' as first function

`f(x) = [t * (- cos t)]_0^x - int_0^x 1 * (- cos t)  dt`

= `- x cos x + [sin t]_0^x`

= `- x cos x + sin x`

`f^1(x) = - [1 * cos x - x sin x] + cos x`

= `x sin x`