Question

\( \displaystyle \int \frac{dx}{\cos^{2/3}x\ \sin^{4/3}x} \) is

• A. \( 3\tan^{3}x + C \)
• B. \( 3\tan^{1/3}x + C \)
• C. \( -3\tan^{1/3}x + C \)
• D. \( -3\tan^{-1/3}x + C \)
• E. \( 3\tan^{-1/3}x + C \)

Hint:

When powers of \( \sin x \) and \( \cos x \) look complicated, try the substitution \( \tan x=t \). It often converts the integrand into a simple power of \( t \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This integral involves powers of sine and cosine in the denominator. A common technique for integrals of the form \( \int \frac{dx}{\sin^m x \cos^n x} \) where \( m+n \) is an even integer is to divide the numerator and denominator by \( \cos^{m+n} x \) to convert the integrand into terms of \( \tan x \) and \( \sec^2 x \).
Step 2: Key Formula or Approach:
1. Check the sum of powers: \( m = 4/3, n = 2/3 \). The sum is \( 4/3 + 2/3 = 6/3 = 2 \), which is an even integer.
2. Divide the numerator and denominator by \( \cos^2 x \).
3. Use the substitution \( u = \tan x \).
Step 3: Detailed Explanation:
The integral is \( I = \int \frac{dx}{\cos^{2/3} x \sin^{4/3} x} \).
Divide the numerator and denominator by \( \cos^2 x \):
\[ I = \int \frac{\frac{1}{\cos^2 x} dx}{\frac{\cos^{2/3} x \sin^{4/3} x}{\cos^2 x}} \] The numerator becomes \( \sec^2 x dx \).
The denominator becomes:
\[ \frac{\sin^{4/3} x}{\cos^{2 - 2/3} x} = \frac{\sin^{4/3} x}{\cos^{4/3} x} = \left(\frac{\sin x}{\cos x}\right)^{4/3} = \tan^{4/3} x \] So the integral transforms to:
\[ I = \int \frac{\sec^2 x}{\tan^{4/3} x} dx = \int (\tan x)^{-4/3} \sec^2 x dx \] Now, let's use the substitution \( u = \tan x \).
Then \( du = \sec^2 x dx \).
Substituting these into the integral:
\[ I = \int u^{-4/3} du \] Using the power rule for integration, \( \int u^n du = \frac{u^{n+1}}{n+1} + C \):
\[ I = \frac{u^{-4/3 + 1}}{-4/3 + 1} + C = \frac{u^{-1/3}}{-1/3} + C = -3u^{-1/3} + C \] Finally, substitute back \( u = \tan x \):
\[ I = -3(\tan x)^{-1/3} + C = -3\tan^{-1/3}x + C \] Step 4: Final Answer:
The integral is equal to \( -3\tan^{-1/3}x + C \). This corresponds to option (D).