Find the indefinite integral.
∫xe^-4xdx
use integration by parts. Let
u = x, du=dx
v = e^-4x dx, v = -1/4 e^-4x
∫ udv = uv - ∫v du
so,
∫(x)(e^-4x dx0 = -1/4 xe^-4x + 1/4 ∫e^-4x dx
= -1/4 xe^-4x - 1/16 e^-4x
= -1/16 e^-4x (4x+1) + C
To find the indefinite integral of ∫xe^(-4x)dx, we can use the technique of integration by parts.
Integration by parts is based on the product rule of differentiation, which states that if we have two functions u(x) and v(x), the derivative of their product is given by:
d/dx(u(x)v(x)) = u(x) * d/dx(v(x)) + v(x) * d/dx(u(x))
The integration by parts formula is expressed as:
∫u(x)v'(x)dx = u(x)v(x) - ∫v(x)u'(x)dx
In this case, we can assign u(x) = x and v'(x) = e^(-4x). Now, we need to calculate u'(x) and v(x):
Taking the derivative of u(x) = x, we get:
u'(x) = 1
For v'(x) = e^(-4x), we can integrate it to find v(x):
∫e^(-4x)dx = -(1/4)e^(-4x)
Now, let's use the integration by parts formula:
∫x*e^(-4x)dx = u(x)v(x) - ∫v(x)u'(x)dx
= x * (-(1/4)e^(-4x)) - ∫(-(1/4)e^(-4x))(1)dx
= -(1/4)xe^(-4x) + (1/4)∫e^(-4x)dx
Now, we can evaluate the integral on the right side:
∫e^(-4x)dx = -(1/4)e^(-4x)
Substituting this result back into the equation:
∫x*e^(-4x)dx = -(1/4)xe^(-4x) + (1/4) * (-(1/4)e^(-4x)) + C
= -(1/4)xe^(-4x) - (1/16)e^(-4x) + C
Therefore, the indefinite integral of ∫xe^(-4x)dx is -(1/4)xe^(-4x) - (1/16)e^(-4x) + C, where C is the constant of integration.