Solve this integral step by step?

Question

The problem is..... cos(sqrt[x])/sqrt[x]

I have tools that can show me the final answer, but I would like to know how it is done step by step. I dont see the point in knowing the answer without actually knowing how to get there.

Answer 1

∫cos(√x) / √x dx
Let u = √x so x = u² or dx = 2udu
So the integral becomes
∫cos(u) / u 2udu
=2∫cos(u) du
=2sin(u) + c
=2sin(√x) + c

Answer 2

Use substitution

Let u = sqrt x
then du = 1/2 x ^-1/2 dx
So, 2du = x ^ -1/2 dx

Substituting u in original
integral cos (sqrt x) / sqrt x dx
integral 2 * cos u du

integrate --> 2 sin u

--> substitute back for u
2 sin u = 2 sin (sqrt x) + c

and that's the answer 2 sin (sqrt x) + c

Answer 3

let u = sqrt x, then du = (1/(2 sqrt x)) dx, or dx/sqrt x = 2 du,
so:

integral((cos sqrt x)/sqrt x) dx = integral(2 cos u du)
= 2 sin u + C
= 2 sin sqrt x + C.

Answer 4

I'll use S for the integration symbol.

S cos(sqrt[x])/sqrt[x] dx

Let u = sqrt(x)

du = 1/2 x^{-1/2}dx = 1/(2sqrt(x))dx

Thus 2du = dx/sqrt(x)

We can change this to

S cos(u) * 2du

= 2 S cos(u) * du
= 2sin(u) + c

Now switch back to x's
Since u = sqrt(x)

S cos(sqrt[x])/sqrt[x] dx = 2sin(sqrt(x)) + c

Answer 5

∫(x - 1)/[x√(x + 2)] dx ∫1/√(x + 2) dx - ∫dx/(x)(√(x + 2) For the remaining integral, let u = √(x + 2). Then it follows that u² - 2 = x. 2u du = dx 2√(x + 2) - ∫2u du/(u)(u² - 2) 2√(x + 2) - ∫2/(u² - 2) du 2√(x + 2) - ∫2/(u + √2)(u - √2) du 2√(x + 2) - 1/√2 * ∫1/(u - √2) - 1/(u + √2) du 2√(x + 2) - 1/√2*ln|(√(x + 2) - √2)/(√(x + 2) + √2)| + C

Answer 6

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