Integral problem help!!!!!!!!!!!!?

Question

Is there any way to solve the 1/x ∫ x^2 sin (4.156/x) dx without using a calculator? I think you can pull out the x^2 from the integral to make it x ∫ sin(4.156/x) but I don't know how to go from there. Help is greatly appreciated.

Answer 1

First of all, you CANNOT pull out the x² from the
integral. You can only pull out constants, never
expressions involving variables.
So, let's let a = 4.156
and look at
∫ x² sin(a/x) dx
Let's let u = a/x
x = a/u
dx = -a/u²
x² = a²/u².
Then we get
∫ -a³/u^4 * sin(u) du,
which is
-a^3∫ sin(u)/u^4 du. (*)
Unfortunately, this last integral is
not elementary.
The integral of sin(u)/ u^k is always nonelementary
for k > 0.
(*) can be reduced to ∫ cos(x)/ x dx by repeated
integration by parts.
which is nonelementary.

Answer 2

you can´t pull out x^2.Try 4.156/x =z so -4.156/x^2 dx = dz
so x^2 = 4.156^2/z^2 and the integral becomes
K*1/x Int (sinz)/z^4 dz which can´be expressed by known elementary functions