Question:
Evaluate the integral
\[
\int \frac{x^4 + 1}{x^6 + 1} dx.
\]
Evaluate the integral
\[
\int \frac{x^4 + 1}{x^6 + 1} dx.
\]
Show Hint
For rational functions, factor and use partial fraction decomposition.
Updated On: Mar 18, 2026
- \( \tan^{-1} x - \tan^{-1} x^3 + c \)
- \( \tan^{-1} x - \frac{1}{3} \tan^{-1} x^3 + c \)
- \( \tan^{-1} x + \tan^{-1} x^3 + c \)
- \( \tan^{-1} x + \frac{1}{3} \tan^{-1} x^3 + c \)
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Splitting the fraction
\[
I = \int \frac{x^4 + 1}{x^6 + 1} dx.
\]
Factoring:
\[
x^6 + 1 = (x^2 + 1)(x^4 - x^2 + 1).
\]
Step 2: Using partial fractions
\[
\frac{x^4 + 1}{x^6 + 1} = \frac{A}{x^2 + 1} + \frac{B}{x^4 - x^2 + 1}.
\]
Solving for coefficients and integrating leads to:
\[
I = \tan^{-1} x + \frac{1}{3} \tan^{-1} x^3 + c.
\]
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