Question

\( \int \frac{x}{1+x^4} dx = \)}

• A. \( \frac{1}{2} \tan^{-1} x^2 + c \)
• B. \( 2 \tan^{-1} x + c \)
• C. \( \frac{1}{2} \tan^{-1} x + c \)
• D. \( \tan^{-1} x^2 + c \)

Hint:

Whenever the numerator power is $(n-1)$ and denominator power is $(2n)$, substitute $u = x^n$.

Answer 1

The Correct Option is A

Step 1: Substitution
Let $u = x^2 \implies du = 2x dx \implies x dx = du/2$.
Step 2: Integral Setup
$\int \frac{1/2}{1 + u^2} du = \frac{1}{2} \int \frac{1}{1+u^2} du$.
Step 3: Integration
$\frac{1}{2} \tan^{-1} u + c$.
Step 4: Back Substitution
$\frac{1}{2} \tan^{-1} x^2 + c$.
Final Answer:(A)