What is the antiderivative of the followring expression?
x^(-1/2) sin(2x^(-3/2))
After trying to figure out this problem, I have the suspicion that the antiderivative cannot be found using substition method, would this assumption be correct?
no idea
Please, if you don't have anything helpful to contribute, save both of us some time. Please, this problem is really getting to me and I don't want any jokes or non-serious answers, thank you.
ok im sorry :(
Thank you for the apology, no hard feelings!
I too have messed around with this a bit, and can't see to get anywhere
Tried integration by parts, only got worse and worse.
I sent it through the Wolfram integrator , and it came up with terrible looking answer containing complex numbers.
http://integrals.wolfram.com/index.jsp?expr=x%5E%28-1%2F2%29+sin%282x%5E%28-3%2F2%29%29+&random=false
What level is this?
Are you sure there is no typo?
To find the antiderivative of the given expression, we can try using the method of substitution. However, in this case, the expression involves a combination of a power of x and a trigonometric function, which makes it difficult to find a suitable substitution.
You are correct to suspect that substitution may not be the most straightforward method to find the antiderivative in this case. There are several techniques for evaluating integrals, and sometimes it may require a combination of methods to find the solution.
One alternative approach you can try is integration by parts. Integration by parts is a method based on the product rule for derivatives. The formula for integration by parts is:
∫ u dv = uv - ∫ v du
Let's assign u and dv to our expression:
u = x^(-1/2) (differentiate u to find du)
dv = sin(2x^(-3/2)) dx (integrate dv to find v)
Differentiating u:
du/dx = (-1/2) x^(-3/2) dx
Integrating dv:
∫ sin(2x^(-3/2)) dx
Here, we encounter another challenging integral involving the sine function and a fractional power of x. At this point, integration by parts may not lead to a straightforward solution either.
In such cases, some integrals can be evaluated using special functions, like the error function (erf) or Bessel functions. However, for the given expression, trying to find an explicit antiderivative using elementary functions might not be possible.
Therefore, your suspicion is correct - it appears that the antiderivative of the expression x^(-1/2) sin(2x^(-3/2)) cannot be found using basic techniques of integration.