The integral ∫ sec^(2/3) x csc^(4/3) x dx is equal to (Hence C is a constant of integration) (1) 3ta

Q: Wait, why did we multiply the denominator by $\cos^{4/3} x$ specifically? Why not $\cos^{2/3} x$?

That is a fantastic question! We chose $\cos^{4/3} x$ because our goal is to create a $\tan x$ term, which is $\frac{\sin x}{\cos x}$. We already had $\sin^{4/3} x$ in the denominator. To turn that into $\tan^{4/3} x$, we needed to divide it by $\cos^{4/3} x$. By multiplying and dividing by $\cos^{4/3} x$, we effectively 'forced' the denominator to contain the $\tan$ term while leaving the remaining $\cos$ terms to combine into $\cos^2 x$, which is the perfect partner for our $\sec^2 x$ differential.

Q: Can't we just use the identity $\sin^2 x + \cos^2 x = 1$ to simplify this?

While that identity is the backbone of trigonometry, applying it here would actually make the expression more complicated. The power of the substitution method lies in recognizing the derivative of $\tan x$ (which is $\sec^2 x$) hidden within the expression. If you try to use $\sin^2 x + \cos^2 x = 1$, you will likely end up with a mix of terms that don't easily cancel out. Always look for the 'derivative pair' first—it is the most efficient path to the solution.

Q: Why is the final answer negative? Did I miss a sign somewhere?

You didn't miss a sign! The negative sign is a natural result of the integration process. When we integrated $u^{-4/3}$, we added $1$ to the exponent, resulting in $-1/3$. Dividing by this new exponent ($-1/3$) is equivalent to multiplying by $-3$. That negative coefficient is mathematically correct and is a common feature in integrals involving reciprocal powers. Trust your algebra!

Question

The integral $\int \sec^{2/3} x \csc^{4/3} x dx$ is equal to (Hence C is a constant of integration)

Options:

(A) $3	an^{-1/3} x + C$

(B) $\frac{3}{4} 	an^{4/3} x + C$

(C) $-3\cot^{-1/3} x + C$

(D) $-3	an^{-1/3} x + C$

Sigma Percentile · Accepted Answer

### The Art of the Trigonometric Transformation

Welcome, fellow traveler on the JEE journey. Today, we are going to dismantle a problem that, at first glance, looks like a tangled mess of fractional powers: the integral of $\sec^{2/3} x \csc^{4/3} x \, dx$.

I know exactly what you are thinking. "Fractional powers? Secant and Cosecant mixed together? How on earth do I integrate this?"

Take a deep breath. In the world of JEE Advanced, intimidation is often the first layer of the problem. Once we peel that back, we find a beautiful, elegant structure waiting for us.

### Phase 1: The Sine-Cosine Foundation

Our first instinct should always be to return to the basics. Trigonometry is a language, and $\sec x$ and $\csc x$ are just dialects of $\sin x$ and $\cos x$.

Let us translate the problem into a more familiar tongue. We know that $\sec x = \frac{1}{\cos x}$ and $\csc x = \frac{1}{\sin x}$. Substituting these into our integral, we get:

$$I = \int \frac{1}{\cos^{2/3} x \sin^{4/3} x} \, dx$$

Now, look at this expression. It is much cleaner, isn't it? We have moved from a product of two complex functions to a single fraction.

But we still have a problem: we have two different trigonometric functions in the denominator, and neither is the derivative of the other. We need a strategy.

### Phase 2: The 'Sum of Powers' Secret

Here is the "Aha!" moment that separates the masters from the novices. Look at the exponents in the denominator: $\frac{2}{3}$ and $\frac{4}{3}$.

What happens when we add them? $\frac{2}{3} + \frac{4}{3} = \frac{6}{3} = 2$. The sum is an even integer!

In the JEE playbook, whenever the sum of the powers of $\sin x$ and $\cos x$ in the denominator is an even integer, it is a massive signal to convert the expression into $	an x$ and $\sec^2 x$. This is the key that unlocks the door.

### Phase 3: The Algebraic Dance

To force this conversion, we need to create a $	an x$ term. We have $\sin^{4/3} x$ in the denominator. If we divide it by $\cos^{4/3} x$, we get $	an^{4/3} x$.

But we cannot just divide by $\cos^{4/3} x$ without multiplying by it as well. So, we perform the following manipulation:

$$I = \int \frac{1}{\cos^{2/3} x \sin^{4/3} x \cdot \frac{\cos^{4/3} x}{\cos^{4/3} x}} \, dx$$

Now, let us group the terms with precision. We pair the $\sin^{4/3} x$ with the $\cos^{4/3} x$ to form $	an^{4/3} x$.

What remains? We are left with $\cos^{2/3} x \cdot \cos^{4/3} x$. When we multiply these, we add the exponents: $\frac{2}{3} + \frac{4}{3} = 2$.

Thus, we are left with $\cos^2 x$. Our integral now looks like this:

$$I = \int \frac{1}{	an^{4/3} x \cdot \cos^2 x} \, dx$$

And here is the magic: $\frac{1}{\cos^2 x}$ is simply $\sec^2 x$. Our integral transforms into:

$$I = \int \frac{\sec^2 x}{	an^{4/3} x} \, dx$$

### Phase 4: The Substitution

Do you see it now? The derivative of $	an x$ is $\sec^2 x$. It is sitting right there in the numerator, waiting for us to use it.

Let us perform a substitution: let $u = 	an x$. Then, $du = \sec^2 x \, dx$. The entire integral collapses into a simple power rule problem:

$$I = \int \frac{1}{u^{4/3}} \, du = \int u^{-4/3} \, du$$

Applying the power rule, $\int u^n \, du = \frac{u^{n+1}}{n+1} + C$, we get:

$$I = \frac{u^{-4/3 + 1}}{-4/3 + 1} + C = \frac{u^{-1/3}}{-1/3} + C$$

### Phase 5: The Final Flourish

Simplifying the coefficient, we get $-3 u^{-1/3} + C$. Finally, we substitute back $u = 	an x$ to return to our original variable.

The result is:

$$I = -3 	an^{-1/3} x + C$$

Look at that. We started with a terrifying expression, and through logical, step-by-step manipulation, we arrived at a clean, elegant solution.

This is the beauty of calculus. It is not about memorizing formulas; it is about recognizing patterns and having the confidence to manipulate them. You have mastered this technique today. Keep this "sum of powers" trick in your toolkit—it will serve you well in many more problems to come.

Animated Solution for Mathematics - Indefinite Integration: The integral $\int sec^{2/3} x csc^{4/3} x d x$ is equal to (Hence C is a constant of integration)

Select Answer:

Visualized Solution

Analyze the Given Integral

Convert to Sine and Cosine

The Tangent Transformation Strategy

Apply the Manipulation

Forming the Tangent Term

Simplifying the Cosine Powers

Introducing Secant Squared

The Substitution Step

Rewrite the Integral in terms of $u$

Apply the Power Rule

Simplify the Exponent

Simplify the Coefficient

Back Substitution and Final Result

The Sigma Insight: Integration by Substitution

The Art of the Trigonometric Transformation

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

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Animated Solution for Mathematics - Indefinite Integration: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The integral ∫sec2/3xcsc4/3xdx is equal to (Hence C is a constant of integration)