lim as b approaches infinity of integral of xe^-x^2 from 0 to b
aka
lim b->inf int(0,b) x*e^(-x^2) dx
To find the limit of the integral as b approaches infinity, we can evaluate the integral first and then take the limit.
Let's evaluate the integral of xe^(-x^2) with respect to x from 0 to b:
∫(0 to b) x*e^(-x^2) dx
We can solve this integral by using a simple u-substitution. Let u = -x^2, then du = -2x dx. Rearranging, we have dx = (-1/2) * du.
Substituting the values into the integral, we get:
∫(-b^2 to 0) (-1/2) * e^u du
Next, we can simplify this expression by pulling out the constant -1/2 and changing the limits from -b^2 to 0 to 0 to b^2:
-(1/2) * ∫(0 to b^2) e^u du
Using the fundamental theorem of calculus, we can evaluate this integral:
-(1/2) * [e^u] from 0 to b^2
Substituting the upper and lower limits into the expression, we get:
-(1/2) * (e^(b^2) - e^0)
Since e^0 is equal to 1, we can simplify further:
-(1/2) * (e^(b^2) - 1)
Now, let's take the limit of this expression as b approaches infinity:
lim (b->∞) [-(1/2) * (e^(b^2) - 1)]
To find the limit, we need to consider the behavior of e^(b^2) as b approaches infinity. As b grows larger, b^2 will also increase, causing e^(b^2) to grow exponentially. So, the limit of e^(b^2) as b approaches infinity is infinity.
Therefore, applying this to our expression, we have:
lim (b->∞) [-(1/2) * (e^(b^2) - 1)] = -(1/2) * (∞ - 1) = ∞
Hence, the limit of the integral as b approaches infinity is infinity.