Evaluate the following :∫cos √x dx

1.	Evaluate the following :∫cos √x dx
Answer» Let I = ∫cos √x dx Put √x = t ∴ x = t² ∴ dx = 2t dt ∴ I = ∫(cos t) 2t dt = ∫ 2t cos t dt = 2t ∫cos t dt – ∫[d/dt(2t) ∫cos t dt]dt = 2t sin t – ∫2 sin t dt = 2t sin t + 2 cos t + c = 2[√x sin√x + cos√x] + c.

Answer»

Let I = ∫cos √x dx

Put √x = t

∴ x = t²

∴ dx = 2t dt

∴ I = ∫(cos t) 2t dt

= ∫ 2t cos t dt

= 2t ∫cos t dt – ∫[d/dt(2t) ∫cos t dt]dt

= 2t sin t – ∫2 sin t dt

= 2t sin t + 2 cos t + c

= 2[√x sin√x + cos√x] + c.

Discussion