integrate -2/xln^4(x)...plz help me..give me an idea on how to start..plz
The derivative of the ln(x) function is 1/x and this is multiplying the
ln^4(x). You can thus write the integral as:
-2 * 1/5 ln^5(x) + constant.
is that the answer?
wut about the x infront of ln^(x)...becasue the orignial question is -2/xln^4(x)...wut happens to da x?
I read that as -2 divided by x. If you differentiate Ln^5(x) you get 5 Ln^4(x) times 1/x.
why should i differentiat it if i am trying to integrate it?
can i also use integration by parts to solve this?
I apologize for the confusion. You are correct, we are trying to integrate the function, not differentiate it.
To integrate -2/xln^4(x), we can use integration by parts.
Integration by parts is a technique based on the product rule for differentiation. The product rule states that if you have two functions u(x) and v(x), then the derivative of their product, uv, is given by u'v + uv'.
In this case, we can choose u = ln^4(x) and dv = -2/x dx.
To find du, we differentiate u with respect to x:
du = (1/x) * 4ln^3(x) dx = 4ln^3(x)/x dx
To find v, we integrate dv with respect to x:
v = ∫ (-2/x) dx = -2ln(abs(x))
Now, we can apply the integration by parts formula:
∫ -2/xln^4(x) dx = uv - ∫ v du
Substituting the values we know:
∫ -2/xln^4(x) dx = ln^4(x)(-2ln(abs(x))) - ∫ (-2ln(abs(x)))(4ln^3(x)/x) dx
Simplifying, we get:
∫ -2/xln^4(x) dx = -2ln^5(x) + 8∫ ln^3(x)/x dx
We can continue integrating by parts on the remaining integral until we reach a point where it's easy to solve.