Question

If 'n' is a natural number, then $\int \frac{\sin^n x}{\cos^{n+2} x} dx =$

• A. $\frac{\tan^{n-1} x}{n - 1} + C$
• B. $\frac{\tan^n x}{n} + C$
• C. $\frac{\tan^{n+2} x}{n + 2} + C$
• D. $\frac{\tan^{n+1} x}{n + 1} + C$

Hint:

When you see an integral with $\sin^m x$ and $\cos^p x$ in the numerator and denominator, check the difference in their powers. If the power of cosine in the denominator is 2 greater than the power of sine in the numerator, separating out a $\frac{1}{\cos^2 x}$ will always cleanly produce a $\tan$ and $\sec^2$ pair for an easy u-substitution.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The integral involves a fraction with powers of sine and cosine. The goal is to manipulate the integrand algebraically into a recognizable form that allows for a standard substitution, typically involving tangent and secant functions.

Step 2: Key Formula or Approach:
Rewrite the denominator $\cos^{n+2} x$ by splitting it into $\cos^n x \cdot \cos^2 x$. This allows the formation of $\tan^n x$ and $\sec^2 x$, setting up a perfect u-substitution where $u = \tan x$ and $du = \sec^2 x dx$.

Step 3: Detailed Explanation:
The given integral is: \[ I = \int \frac{\sin^n x}{\cos^{n+2} x} dx \] Let's decompose the denominator: \[ I = \int \frac{\sin^n x}{\cos^n x \cdot \cos^2 x} dx \] We can group the terms with the power $n$ together: \[ I = \int \left( \frac{\sin x}{\cos x} \right)^n \cdot \frac{1}{\cos^2 x} dx \] Using basic trigonometric identities ($\frac{\sin x}{\cos x} = \tan x$ and $\frac{1}{\cos^2 x} = \sec^2 x$): \[ I = \int (\tan x)^n \cdot \sec^2 x dx \] Now the integral is in a standard form for substitution. Let: \[ u = \tan x \] Then the differential is: \[ du = \sec^2 x dx \] Substitute $u$ and $du$ into the integral: \[ I = \int u^n du \] Using the power rule for integration $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (since n is a natural number, $n \neq -1$): \[ I = \frac{u^{n+1}}{n+1} + C \] Finally, substitute back $u = \tan x$: \[ I = \frac{\tan^{n+1} x}{n+1} + C \]

Step 4: Final Answer:
The integral evaluates to $\frac{\tan^{n+1} x}{n+1} + C$.