intergral cos(x^1/2)?

Answer 1

Integral ( cos(x^(1/2)) dx )

To solve this is a two step process. First, we're going to use substitution.

Let z = x^(1/2). Square both sides,
z^2 = x. Therefore

2z dz = dx

So our integral becomes

Integral ( cos(z) 2z dz )

Rearranging this and pulling out the constant 2,

2 * Integral (z cos(z) dz)

Now, we have to use integration by parts.

Let u = z. dv = cos(z) dz.
du = dz. v = sin(z)

2 [ zsin(z) - Integral (sin(z) dz) ]

And now, we have an easy integral to solve.

2 [zsin(z) - (-cos(z))] + C

2 [zsin(z) + cos(z)] + C

2z sin(z) + 2cos(z) + C

But z = x^(1/2), so we have

2x^(1/2) sin(x^(1/2)) + 2cos(x^(1/2)) + C

Integral (