Problem 60 - Essential Calculus

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int \frac{d x}{25+x^{2}}=\frac{1}{25} \arctan \frac{x}{25}+C $$

Step-by-Step Solution

Answer

The statement is False. The correct statement is: \(\int \frac{1}{25+x^{2}} dx = \frac{1}{5} \arctan \frac{x}{5} + C\)

Step 1: Analyze the formula

We know the integral of \(\frac{1}{a^{2} + x^{2}}\) is \(\frac{1}{a} \arctan \frac{x}{a} + C\), according to the table of common integrals. So let's see whether the statement matches this formula.

Step 2: Evaluate the function

In this case, \(a^{2}=25\), so \(a=5\). Therefore, if the statement provided is correct, then we should have \(\int \frac{1}{25+x^{2}} dx = \frac{1}{5} \arctan \frac{x}{5} + C\)

Step 3: Compare the calculated result to the given statement

Comparing this result with the given statement \(\int \frac{1}{25+x^{2}} dx = \frac{1}{25} \arctan \frac{x}{25} + C\), we can see that they are not the same.

Step 4: Draw conclusion

The factor is not correctly stated in the given statement. Instead of \(\frac{1}{25}\), it should be \(\frac{1}{5}\), and instead of \(\frac{x}{25}\), it should be \(\frac{x}{5}\). Thus, the statement given in the problem is False.