Problem 35 - Essential Calculus

Question

Determine which of the integrals can be found using the basic integration formulas you have studied so far in the text. (a) \(\int \frac{1}{\sqrt{1-x^{2}}} d x\) (b) \(\int \frac{x}{\sqrt{1-x^{2}}} d x\) (c) \(\int \frac{1}{x \sqrt{1-x^{2}}} d x\)

Step-by-Step Solution

Answer

The integrals in (a) can be found using the basic integration formulas studied. The integral in (b) can potentially be solved depending on interpretation of 'basic'. The integral (c) does not match any basic formula and therefore cannot be solved using basic integration formulas.

Step 1: Analyze each integral

For each given integral, inspect the function that's being integrated (the integrand), and compare it with known basic integration formulas.

Step 2: Identify a match for (a)

Looking at the integral \(\int \frac{1}{\sqrt{1-x^{2}}} d x\), it is noticed that this matches the integral of the secant function, which is known to be \(\int \sec(x)dx = \ln |\sec(x) + \tan(x)| + c\). Therefore, the given integral can be found using basic integration formulas.

Step 3: Identify a match for (b)

The integral \(\int \frac{x}{\sqrt{1-x^{2}}} d x\) cannot be directly matched with a basic integration formula. However, it can be solved using the method of substitution, which might not be considered as 'basic' depending on interpretation. Hence, it is uncertain whether part (b) satisfies the condition.

Step 4: Identify a match for (c)

The integral \(\int \frac{1}{x \sqrt{1-x^{2}}} d x\) doesn't match any of the basic integration formulas. Therefore, the integral in part (c) cannot be found using basic integration formulas.