Integration by Parts Question?

Question

Hi, could anyone help me out with this problem?

Make a substitution first, then use integration by parts to evaluate.

fINT( sin (sqrt(x)) dx )

Thanks for any help!

Answer 1

∫ sin(√x) dx =

let √x = t →

x = t² →

dx = 2t dt

then substituting, you get:

∫ sin(√x) dx = ∫ sin t 2t dt =

2 ∫ t sin t dt =

thus, integrating it by parts, let:

t = u → dt = du

sin t dt = dv → - cos t = v

then:

∫ u dv = u v - ∫ v du →

2 ∫ t sin t dt = 2 [- t cos t - ∫ (- cos t) dt] =

2 (- t cos t + ∫ cos t dt) =

- 2t cos t + 2 ∫ cos t dt =

- 2t cos t + 2sin t + C

finally, subsitute back t = √x, yielding:

∫ sin(√x) dx = - 2√x cos(√x) + 2sin(√x) + C

I hope it helps..

Bye!

Answer 2

I got wierd answers....give me an hour to think