Problem 16 - Essential Calculus

Question

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int \frac{x^{3} e^{x^{2}}}{\left(x^{2}+1\right)^{2}} d x $$

Step-by-Step Solution

Answer

The solution to the integral \(\int \frac{x^{3} e^{x^{2}}}{\left(x^{2}+1\right)^{2}} d x\) is -\(\frac{e^{x^{2}}}{2(x^{2}+1)}\) + \(\frac{e^{x^{2}}}{2}\).

Step 1 - Make a Suitable Substitution

Let's substitute \(x^{2}\) with \(u\), which simplifies the expression in the integral. It implies that \(2xdx = du\), or in other words \(dx = du/(2x)\).

Step 2 - Change the Variable in the Integral

Now, replace \(x^{2}\) in the integral with \(u\) and \(dx\) with \(du/(2x)\). The integral then becomes: \(\int \frac{x^{2}e^u}{{(u+1)}^{2}} \cdot du/(2x)\). The \(x\) in the numerator and the denominator cancel each other out, simplifying the integral to: \(\frac{1}{2} \int \frac{u e^{u}}{(u+1)^{2}} du\).

Step 3 - Solve the Simplified Integral

This integral is more straightforward to solve and can be solved using the method of integration by parts (if necessary). However, in this case it is not required as integration of the function can be solved using direct integration method. \(\frac{1}{2} \int \frac{u e^{u}}{(u+1)^{2}} du\) = -\(\frac{e^{u}}{2(u+1)}\) + \(\frac{e^{u}}{2}\).\)

Step 4 - Substitute Back the Original Variable

To get the final answer, we need to substitute \(u\) back with \(x^{2}\). This gives us the final result of -\(\frac{e^{x^{2}}}{2(x^{2}+1)}\) + \(\frac{e^{x^{2}}}{2}\)