Animated Solution for Mathematics - Indefinite Integration: The value of the integral $\int \frac{\sin \theta -\sin 2\theta ({\sin }^3\theta +{\sin }^2\theta +\sin \theta )}{\sqrt{2{\sin }^4\theta +3{\sin }^2\theta +6}}\frac{d\theta }{1-\cos 2\theta }$ is (where $c$ is a constant of integration)

The integral provided is:

At first glance, this expression appears daunting due to the trigonometric complexity. However, in JEE Advanced, the provided options often serve as a roadmap. Since every option is in the form $\frac{1}{18}[f(\theta ){]}^{\frac{3}{2}}+c$, we can infer that the integral is a disguised version of the power rule:

We extract the core polynomial from the structure of the options. Let us define:

If our hypothesis is correct, the derivative of $z$ must be hidden within the numerator of the integral. We differentiate $z$ with respect to $\theta$ using the chain rule:

Factoring out $12\cos \theta$, we obtain:

This confirms that the complexity of the original integral was merely a mask. By identifying $z$ from the options, we bypass tedious simplification and reduce the integral to:

Applying the power rule $\int z^{\frac{1}{2}}dz=\frac{z^{\frac{3}{2}}}{\frac{3}{2}}$, and multiplying by our constant $\frac{1}{12}$, we get:

Substituting our original definition of $z$ back into the expression, we have:

To match the specific form of the provided options, we use the identity ${\sin }^2\theta =1-{\cos }^2\theta$. Substituting this into $z$ and expanding the terms, we arrive at:

This result matches the required form perfectly. By using the options as a strategic map, we have successfully solved the problem. The final answer is derived from the substitution of this polynomial into the power rule result.