USE INTEGRATION BY PARTS TO FIND EACH INTEGRAL
ƪxe^2x dx
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let u = x
du/dx = 1
du = dx
let dv = e^(2x)dx
dv/dx = e^(2x)
v = (1/2)e^(2x)
formula:
[int] u dv = uv - [int] v du
[int] x e^(2x) dx = (x)(1/2)e^(2x) - [int] (1/2)e^(2x) dx
= (1/2)xe^(2x) - (1/4)e^(2x)
To find the integral of xe^2x, we will use the method of integration by parts. The formula for integration by parts is:
∫u * dv = uv - ∫v * du
Let's assign u = x and dv = e^2x dx. We can then differentiate u to find du, and integrate dv to find v.
Differentiating u:
du = dx
Integrating dv:
∫e^2x dx
To integrate e^2x, we can use the formula for the integral of e^x, which is ∫e^x dx = e^x + C. Therefore, integrating e^2x gives us:
v = (1/2) * ∫e^2x dx = (1/2) * (1/2) * e^2x + C = (1/4) * e^2x + C
Now, we have all the components needed to apply the integration by parts formula:
∫xe^2x dx = uv - ∫v * du
∫xe^2x dx = x * (1/4) * e^2x - ∫(1/4) * e^2x dx
Simplifying, we have:
∫xe^2x dx = (1/4) * x * e^2x - (1/4) * ∫e^2x dx
The remaining integral on the right side is the same as before, and we can substitute it back in:
∫xe^2x dx = (1/4) * x * e^2x - (1/4) * ((1/4) * e^2x + C)
Simplifying further, we get:
∫xe^2x dx = (1/4) * x * e^2x - (1/16) * e^2x - C
Thus, the integral of xe^2x is:
∫xe^2x dx = (1/4) * x * e^2x - (1/16) * e^2x + C, where C is the constant of integration.