Intergrate (sin(x^(1/2))

The problems tell me to do a subsition then use by parts to solve it.
 PhysOrg.com science news on PhysOrg.com >> City-life changes blackbird personalities, study shows>> Origins of 'The Hoff' crab revealed (w/ Video)>> Older males make better fathers: Mature male beetles work harder, care less about female infidelity
 Recognitions: Gold Member Homework Help Science Advisor Set $$u=\sqrt{x}$$
 Recognitions: Homework Help Science Advisor guessing and playing around is also a useful technique sometimes. just looking at that i would try xsin(x^(1/2)) just to see what comes out. well i got some stuff with x^(1/2) in front, which led me to try x^(1/2)cos(x^(1/2)). that came close enough to guess the rest. guessing and playing around is actually easier and faster sometiems than keeping track of all the products and signs in the "parts" algorithm.

Recognitions:
Gold Member
Homework Help

 With questions which tell you to use a certain method I think it is best to give it a go even if you do not see where it might lead. $$\int {\sin \left( {\sqrt x } \right)} dx$$ Let $$u = \sqrt x \Rightarrow \frac{{du}}{{dx}} = \frac{1}{{2\sqrt x }} \Rightarrow dx = 2\sqrt x du$$ From the substitution made earlier you can write: $$dx = 2udu$$ So you now have: $$\int {\sin \left( {\sqrt x } \right)} dx$$ $$= \int {\sin \left( u \right)2u} du$$ $$= - 2u\cos \left( u \right) - \int {\left( { - \cos \left( u \right)} \right)} 2du$$ $$= - 2u\cos \left( u \right) + \int {2\cos \left( u \right)} du$$ $$= - 2u\cos \left( u \right) + 2\sin \left( u \right) + c$$ $$= - 2\sqrt x \cos \left( {\sqrt x } \right) + 2\sin \left( {\sqrt x } \right) + c$$