evaluate the integral:
y lny dy
i know it's integration by parts but i get confused once you have to do it the second time
Leibnitz rule (a.k.a. product rule):
d(fg) = f dg + g df
y lny dy = d[y^2/2 ln(y)] - y/2 dy
---->
Integral of y lny dy =
y^2/2 ln(y) - y^2/4 + const.
Instead of partial integration you can use this trick:
Integral of y^a dy = y^(a+1)/(a+1) (1)
Differentiate both sides w.r.t. a:
Integral of y^a Ln(y) dy =
y^(a+1)Ln(y)/(a+1) -y^(a+1)/(a+1)^2 (2)
uUp to an integration constant)
Substitute a = 1 to obtain the answer. Note that the integration constant we could have added to (1) can still depend on the parameter a. So, if you differentiate it you get an integration constant in (2)
as well.
Integral of y lny dy =
y^2/2 ln(y) - y^2/4 + const.
To evaluate the integral of y lny dy, we can use integration by parts.
Let's start by applying the product rule (Leibnitz rule) to the function y lny:
d(fg) = f dg + g df
In this case, let f = ln(y) and g = y. So, df = (1/y)dy and dg = dy.
Now, we can rewrite the integral as:
∫ y lny dy = ∫ (ln(y))(y) dy
Using the product rule, we have:
∫ (ln(y))(y) dy = (1/2)y^2 ln(y) - ∫ (1/2)y^2 (1/y) dy
Simplifying the integral, we get:
∫ (1/2)y^2 (1/y) dy = (1/2)∫ y dy
This is a simple integral that can be solved easily:
∫ y dy = (1/2)y^2 + C
Therefore, the final result of the integral is:
∫ y lny dy = (1/2)y^2 ln(y) - (1/2)y^2 + C
where C is the constant of integration.
To evaluate the integral of y lny dy, we can use the technique of integration by parts. The first step is to identify which function to differentiate and which function to integrate. In this case, we'll let u = ln(y) and dv = y dy.
To find du, we take the derivative of u with respect to y, which gives du = 1/y dy.
To find v, we integrate dv with respect to y, which gives v = y^2/2.
Now we can apply the integration by parts formula:
∫ u dv = uv - ∫ v du
Substituting the values we found:
∫ y lny dy = (ln(y))(y^2/2) - ∫ (y^2/2)(1/y) dy
Simplifying the equation gives:
∫ y lny dy = (y^2/2) ln(y) - ∫ y/2 dy
The integral on the right side, ∫ y/2 dy, is a simple integral that can be solved directly:
∫ y/2 dy = (1/2) * ∫ y dy = (1/2) * (y^2/2) + C1
Where C1 represents the constant of integration.
So, the final result is:
∫ y lny dy = (y^2/2) ln(y) - (y^2/4) + C1
Where C1 is the constant of integration.