If `int x/sqrt(1+x^2+sqrt((1+x^2)^3)dx`= `ksqrt(1+sqrt(1+x^2))+c`. find `k`

Explanation

To solve the integral \( \int \frac{x}{\sqrt{1+x^2+\sqrt{(1+x^2)^3}}} \, dx \) and find the value of \( k \) such that the integral equals \( k \sqrt{1+\sqrt{1+x^2}} + c \), we can follow these steps:

### Step 1: Substitution
Let \( t = 1 + x^2 \). Then, we differentiate both sides:
\[ dt = 2x \, dx \implies x \, dx = \frac{dt}{2} \]

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