Integrate e^(2x)*((2x-1)/(4(x)^2))^2
My thoughts on this question :
I simplfied the terms after the "*" to get three separate integrals :
integrate (e^(2x)) + ((e^(2x))/4(x^2)) + ((e^(2x))/x)
The answer for the first integral is obvious and then I was able to simplify 2nd to the form of 3rd and I don't see a way to integrate the 3rd one. I need help on solving the 3rd integral.
Thanks!
The 3rd one cannot be done using elementary functions.
It involves Ei(x), the exponential integral, which is just defined as
∫ e^x / x dx
Thanks @oobleck.
Is there any other way to solve this using elementary functions without involving the integral e^(x)/x?
Well, I guess you could reduce it to an infinite series and integrate term by term.
Not very satisfying though ...
To solve the third integral, which is ∫ (e^(2x))/x dx, we can use integration by parts.
Integration by parts states that for the product of two functions, u and v, the integral of their product can be expressed as:
∫ u dv = u*v - ∫ v du
In our case, let's choose u = 1/x and dv = e^(2x) dx. Taking the derivative of u, we get du = -1/x^2 dx, and to find v, we integrate dv:
v = ∫ e^(2x) dx = (1/2)e^(2x)
Now, we can apply the integration by parts formula:
∫ (e^(2x))/x dx = u*v - ∫ v du
= (1/x) * (1/2)e^(2x) - ∫ (1/2)e^(2x) * (-1/x^2) dx
Simplifying the equation, we have:
∫ (e^(2x))/x dx = (1/2x)e^(2x) + (1/2) ∫ (e^(2x))/(x^2) dx
Now, we have reduced the integral to a slightly simpler form. The remaining integral, ∫ (e^(2x))/(x^2) dx, is known as an exponential integral and does not have a simple elementary antiderivative.
However, it can be expressed using a special function called the exponential integral (Ei(x)). Therefore, the final integral is:
∫ (e^(2x))/x dx = (1/2x)e^(2x) + (1/2) * Ei(2x) + C
where C is the constant of integration.
So, the solution to the original integral, ∫ e^(2x)*((2x-1)/(4(x)^2))^2 dx, can be expressed as:
∫ e^(2x)*((2x-1)/(4(x)^2))^2 dx = e^(2x) + (1/4) * (1/2x)e^(2x) + (1/4) * (1/2) * Ei(2x) + C
where C is the constant of integration.