How do we integrate xlne^x .
ln(e^x) = x, so x(ln(e^x)) = x^2
So just integrate x^2.
To integrate the function xlne^x, we can use the technique of integration by parts. The formula for integration by parts is:
∫ u dv = uv - ∫ v du
Let's start by choosing u and dv:
u = ln(x) (Choose the more complicated function)
dv = xe^x dx (Choose the remaining part of the function)
Now, we need to find du and v by differentiating u and integrating dv, respectively:
du = (1/x) dx (Differentiate ln(x) with respect to x)
v = ∫ xe^x dx (Integrate xe^x with respect to x)
To find v, we can use integration by parts again:
Let u = x
Therefore, dv = e^x dx
Using the integration by parts formula, we get:
v = ∫ xe^x dx = xe^x - ∫ e^x dx
The integral of e^x is simply e^x, so we have:
v = xe^x - e^x
Now, we can substitute the values of u, v, du, and dv into the formula for integration by parts:
∫ xlne^x dx = uv - ∫ v du
= ln(x)(xe^x - e^x) - ∫ (xe^x - e^x) (1/x) dx
Simplifying this expression gives:
∫ xlne^x dx = ln(x)(xe^x - e^x) - ∫ (e^x - 1) dx
= ln(x)(xe^x - e^x) - (xe^x - e^x) + C
Therefore, the integral of xlne^x is given by ln(x)(xe^x - e^x) - (xe^x - e^x) + C, where C is the constant of integration.