Integral of [sin(sqrt(x))]/sqrt(x)

Let u = sqrt(x) = x^1/2, then du = 1/2 x^-1/2 dx and dx = 2x^1/2 du by rearranging

Substitute u for x^1/2 gives dx = 2u du

integral (sin(sqrt(x))/sqrt(x) dx) = integral (sin(u)/u * 2u du) by substitution

= integral (2 sin u du) = -2 cos u + C = -2 cos(sqrt(x)) + C (substituting back)

∫(sin√x)/(√x) dx = ?

first, set u = √x, we get du = 2dx/(√x), or dx = 1/2√x du

Now

∫(sin√x)/(√x) dx = ∫1/2 (sin u) du
= -1/2 cos u +C
= -1/2 cos√x + C