- #1
bhatnv
- 6
- 0
how would i go about finding the definite integral of this (x^3*e^(x^2))/(x^2+1)^2
Last edited:
bhatnv said:how would i go about finding the definite integral of this [PLAIN]http://www.texify.com/img/%5CLARGE%5C%21f%28x%29%3D%5Cfrac%7Bx%5E3e%5Ex%5E2%7D%7B%28x%5E2%2B1%29%5E2%7D.gif[/QUOTE]
Your integral doesn't show up.
To solve this integral, we can use the substitution method. Let u = x^2+1, then du = 2x dx. Rearranging, we get x dx = du/2. Substituting this into the integral, we get 1/2 * ∫ e^u/ u^2 du. This integral can be solved using integration by parts, and the final result will be 1/2 * e^u/u + C. Substituting back in for u, we get the final answer of 1/2 * e^(x^2+1)/(x^2+1) + C.
The domain of this function is all real numbers except for x = ±i, where i is the imaginary unit. This is because the denominator (x^2+1)^2 cannot be equal to 0, and the exponential function is defined for all real numbers.
No, the integral of x^3 e^(x^2)/(x^2+1)^2 is not an elementary function. It cannot be expressed in terms of elementary functions such as polynomials, exponentials, logarithms, trigonometric functions, etc. This can be seen by trying to integrate it using the standard integration techniques, which will lead to a dead end.
Yes, the integral can be approximated numerically using methods such as the trapezoidal rule or Simpson's rule. These methods use a series of small trapezoids or parabolas to approximate the area under the curve, and as the number of trapezoids or parabolas increases, the approximation becomes more accurate.
This integral has applications in various fields such as physics, engineering, and economics. For example, it can be used to calculate the work done by a variable force, the heat transfer in a system, or the area under a demand curve in economics. It is also used in statistics to calculate probabilities in a normal distribution. In general, integrals are used to model and solve real-world problems that involve continuous change.