Question

Evaluate the indefinite integral: \[ \int \frac{x^3 \sin\left(\tan^{-1}(x^4)\right)}{1+x^8}\, dx \]

• A. \( \frac{1}{4} \cos [\tan^{-1}(x^4)] + C \)
• B. \( -\frac{1}{4} \cos [\tan^{-1}(x^4)] + C \)
• C. \( \frac{1}{4} \sin [\tan^{-1}(x^4)] + C \)
• D. \( -\frac{1}{4} \sin [\tan^{-1}(x^4)] + C \)

Hint:

Whenever you see a composite function like \(f(x^n)\), check if \(x^{n-1}\) appears in the numerator. It is a strong indicator that substitution will simplify the integral immediately.

Answer 1

The Correct Option is B