Question

calculate:

$$ int \frac { 12 x ^ { 2 } } { x ^ { 6 } + 1 } d x = $$

symbolic expression

Explanation:

Step1: Use substitution

Let \( u = x^{3} \), then \( du=3x^{2}dx \), and \( 12x^{2}dx = 4du \).
The integral becomes \( \int\frac{12x^{2}}{x^{6}+1}dx=\int\frac{4du}{u^{2}+1} \).

Step2: Apply integral formula

We know that \( \int\frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}\tan^{- 1}(\frac{x}{a})+C \) (here \( a = 1 \)).
So \( \int\frac{4du}{u^{2}+1}=4\tan^{-1}(u)+C \).

Step3: Substitute back

Since \( u = x^{3} \), the integral \( \int\frac{12x^{2}}{x^{6}+1}dx=4\tan^{-1}(x^{3})+C \).

Answer:

\( 4\tan^{-1}(x^{3})+C \)