Problem 41 - Essential Calculus

Question

Use a computer algebra system to find the integral. Verify the result by differentiation. $$ \int \frac{x^{2}}{\sqrt{x^{2}+10 x+9}} d x $$

Step-by-Step Solution

Answer

The computed integral is \( x\sqrt{x^2 + 10x + 9} - \frac{9}{2}ln|x + \sqrt{x^2 + 10x + 9} + 1| - \frac{5}{2}ln|x + \sqrt{x^2 + 10x + 9} - 1| + C \). After verification by differentiation, it is confirmed that this integral is correct.

Step 1: Compute the integral using computer algebra system

Input the integral into a computer algebra system to calculate it. The integral \(\int \frac{x^{2}}{\sqrt{x^{2}+10 x+9}} d x\) can be computed as \( x\sqrt{x^2 + 10x + 9} - \frac{9}{2}ln|x + \sqrt{x^2 + 10x + 9} + 1| - \frac{5}{2}ln|x + \sqrt{x^2 + 10x + 9} - 1| + C \) where C is the constant of integration.

Step 2: Verify the integral by differentiation

The next part is to verify the result by differentiation. The derivative of the computed integral, using the rule of the chain and the product rule, should return us to the original function \( \frac{x^{2}}{\sqrt{x^{2}+10 x+9}} \).

Step 3: Differentiate the computed integral

Using differentiation rules, compute the derivative of the integral. If the differentiation is done correctly, it should return the function that was integrated. Note that when computing the derivative, the constant of integration 'C' will just disappear as the derivative of a constant is zero.

Step 4: Compare the results

Compare the original function with the derivative result. If they match, then the integral was calculated correctly.