spaniks said:Homework Statement
Use the fact that the integral evaluated from -∞ to +∞ of e^(-x^2) is sqrt(∏) to evaluate the integral from -∞ to +∞ of x^2(e^(-x^2)).
Homework Equations
The Attempt at a Solution
I tried using integration by parts and I came down to an indefinite integral of sqrt(∏)*x^2. I know the answer is sqrt(∏)/2 but I don't see how. Can someone tell me what I am doing wrong please.
The formula for finding the integral of x^2e^(-x^2) is ∫x^2e^(-x^2)dx = -1/2e^(-x^2) + C, where C is the constant of integration.
To solve the integral of x^2e^(-x^2), you can use the substitution method by letting u = -x^2. Then, du/dx = -2x and dx = -du/2x. Substituting these values into the integral, we get ∫x^2e^(-x^2)dx = ∫-(u/2)e^u(-du/2x) = 1/4∫ue^u du. This can be solved using integration by parts or the power rule.
The graph of the integral of x^2e^(-x^2) is a curve that starts at the origin and asymptotically approaches the x-axis on both sides. It has a maximum value of 0.5 at x = 0 and decreases as x increases.
Geometrically, the integral of x^2e^(-x^2) represents the area under the curve y = x^2e^(-x^2) from x = 0 to infinity. This area is bounded by the x-axis and the curve and can be approximated by dividing the area into small rectangles and summing their areas.
The integral of x^2e^(-x^2) has various real-life applications, such as in physics, to calculate the electric field produced by a charged ring, or in statistics, to calculate the probability of a random variable following a normal distribution. It is also used in engineering and economics for modeling and analyzing various systems and processes.