Animated Solution for Mathematics - Indefinite Integration: The value of the integral $\int \frac{{\cos }^{3}x+{\cos }^{5}x}{{\sin }^{2}x+{\sin }^{4}x}dx$ is

The integral we are dissecting is:

When you encounter such a problem in the JEE Advanced, do not panic. Look for the hidden symmetry: the numerator contains odd powers of $\cos x$, while the denominator consists of even powers of $\sin x$.

In the world of integration, odd powers of cosine are a massive hint that a substitution is waiting to happen.

Preparing the Ground

Our first mission is to isolate the differential. We know that if we want to substitute $t=\sin x$, we need a $\cos xdx$ term.

We perform surgery on the numerator by factoring out a single $\cos x$:

Next, we factor the denominator to get ${\sin }^{2}x(1+{\sin }^{2}x)$. Our integral now takes the form:

Converting to Algebraic Form

We must convert all remaining trigonometric terms into $\sin x$ using the identity ${\cos }^{2}x=1-{\sin }^{2}x$. Consequently, ${\cos }^{4}x=(1-{\sin }^{2}x{)}^2$.

Substituting these into the integral, we obtain:

Let $t=\sin x$, which implies $dt=\cos xdx$. The integral transforms into a purely algebraic rational function:

Division and Decomposition

Expanding $(1-t^2{)}^2$ gives $1-2t^2+t^4$. Combining terms, the numerator becomes $t^4-3t^2+2$.

The integral is now:

Since the degree of the numerator equals the degree of the denominator, we must perform polynomial division. We rewrite the numerator as:

Dividing by $t^2(t^2+1)$, we obtain:

We decompose the remainder using partial fractions:

Solving for the coefficients, we find $A=2$ and $B=-6$.

Final Integration

We are left with the simplified integral:

Integrating term by term, we get:

Finally, back-substituting $t=\sin x$, we arrive at the final answer: