Final answer:
To evaluate the indefinite integral of 3x^3/(x^8+1), you can use partial fraction decomposition. Express the rational function as the sum of two fractions, and then apply substitutions and integrate each fraction separately. Combine the results to find the final indefinite integral.
Step-by-step explanation:
To evaluate the indefinite integral of 3x^3/(x^8+1), we can use a technique called partial fraction decomposition. First, we can write the denominator as a product of linear factors: x^8 + 1 = (x^4 + 1)(x^4 - 1). Then, we can express the rational function as the sum of two fractions:
- 3x^3 / (x^4 + 1)
- 3x^3 / (x^4 - 1)
For the first fraction, we can use a substitution u = x^4 + 1 to simplify the expression. For the second fraction, we can use a substitution v = x^4 - 1 to simplify the expression. After applying these substitutions and integrating, we can combine the results to find the indefinite integral.