Animated Solution for Mathematics - Indefinite Integration: For any natural number $m$, evaluate $\int (x^{3m}+x^{2m}+x^m)(2x^{2m}+3x^m+6{)}^{1/m}dx,x>0$

Welcome, fellow traveler on the path to JEE mastery. Today, we are going to dissect an integral that, at first glance, looks like a chaotic mess of powers.

We are evaluating the expression:

It is easy to feel intimidated by the parameter $m$ and the fractional power, but in the world of JEE Advanced, complexity is often just a mask for elegance. Our job is to peel back that mask.

A Hidden Symmetry

Look closely at the powers of $x$. In the first bracket, we have $x^{3m},x^{2m},x^m$. In the second, we have $x^{2m},x^m$.

The powers in the first bracket are exactly $m$ units lower than the powers inside the second bracket if we consider the derivative. This is a breadcrumb trail left by the problem setter. Our goal is to manipulate this expression so that the first bracket becomes the derivative of the second.

Forcing the Match

To make this work, we factor out $x^{m-1}$ from the first bracket. The integral becomes:

Now, we push the $x^{m-1}$ inside the second bracket by utilizing the power $\frac{1}{m}$. We rewrite $x^{m-1}$ as $\frac{1}{m}\cdot \frac{d}{dx}(x^m)$.

Alternatively, we can observe that the derivative of the term $u=2x^{3m}+3x^{2m}+6x^m$ is:

The Collapse of Complexity

Let $u=2x^{3m}+3x^{2m}+6x^m$. Factoring out $6m$, we see:

This matches our first bracket perfectly. We can now rewrite the integral as:

What started as a terrifying algebraic monster has collapsed into a simple power rule integral. We apply the rule $\int u^{n}du=\frac{u^{n+1}}{n+1}+C$.

Performing the integration, we obtain:

Simplifying the exponent and the coefficient:

Substituting back the value of $u$, we arrive at our final answer:

Remember, the key to these problems is always to look for that derivative match. Keep practicing, and you will start to see these patterns everywhere!