Problem 38 - 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) \(\int \frac{1}{1+x^{4}} d x\) (b) \(\int \frac{x}{1+x^{4}} d x\) (c) \(\int \frac{x^{3}}{1+x^{4}} d x\)

Step-by-Step Solution

Answer

Only integral (c)\/ \( \int \frac{x^{3}}{1+x^{4}} d x \) can be solved using the basic integration formulas.

Step 1: Analyzing Integral (a)

Look at integral (a) \( \int \frac{1}{1+x^{4}} d x \). It seems complex as it's not a simple integral and doesn't match any basic integration formulas. Trying substitution or parts method would also be complicated and not straightforward. Hence (a) is not solvable using basic integration formulas.

Step 2: Analyzing Integral (b)

Examine integral (b) \( \int \frac{x}{1+x^{4}} d x \). Let \( u=x^{4}+1 \). Then \( du=4x^3 dx \). But we have \( x \), not \( x^{3} \) in the function, and we cannot make it simple. Therefore, after attempting substitution, it's still found difficult to solve using basic integration formulas.

Step 3: Analyzing Integral (c)

For the integral (c) \( \int \frac{x^{3}}{1+x^{4}} d x \), we can use substitution method which is a basic method for solving Integrals. Here, Let \( u=x^{4}+1 \) Then \( du=4x^{3} dx \). After replacing these values in our original Integral we have \( \frac{1}{4} \int \frac{du}{u} \) which is an elementary logarithmic formula for integration.