evaluate the integral using integration by parts
7xe^-x
u = 7x
du = 7 dx
dv = e^-x dx
v = -e^-x
∫u dv = uv - ∫v du
∫7xe^-x dx = -7xe^-x + 7∫e^-x dx
I think you can take it from there, no?
To evaluate the integral ∫ 7xe^(-x) dx using integration by parts, we can apply the integration by parts formula:
∫ u dv = uv - ∫ v du
Let's consider u and dv as follows:
u = 7x (Take the function whose derivative is easy to compute)
dv = e^(-x) dx (Take the remaining function as dv)
Now, we need to find du and v using the given functions:
du = d(7x) = 7 dx (Take the derivative of u)
To find v, we integrate dv:
∫ e^(-x) dx = -e^(-x) (Integrate the given function)
Now, we have all the necessary components to apply the integration by parts formula:
∫ 7xe^(-x) dx = u*v - ∫ v*du
Plugging in the values:
∫ 7xe^(-x) dx = 7x * (-e^(-x)) - ∫ (-e^(-x)) * 7 dx
Simplifying further:
∫ 7xe^(-x) dx = -7x e^(-x) + 7 * ∫ e^(-x) dx
The integral ∫ e^(-x) dx is a basic integral:
∫ e^(-x) dx = -e^(-x)
Substituting this value back into the equation:
∫ 7xe^(-x) dx = -7x e^(-x) + 7 * (-e^(-x))
Simplifying the equation further:
∫ 7xe^(-x) dx = -7x e^(-x) - 7e^(-x)
Thus, that's the result of the integral using integration by parts.