How do i integrate (cot²x-n+1)/((cosx)^n)

I want to know proper approach of this question

To tackle the integration of the expression \(\frac{\cot^2 x - n + 1}{(\cos x)^n}\), we can break it down into manageable parts. The key is to recognize the relationships between the trigonometric functions involved and how they can simplify the integration process. Let's go through this step by step.

Understanding the Components

The expression consists of two main parts: the numerator \(\cot^2 x - n + 1\) and the denominator \((\cos x)^n\). First, let's rewrite \(\cot^2 x\) in terms of sine and cosine:

• \(\cot^2 x = \frac{\cos^2 x}{\sin^2 x}\)

This gives us:

• \(\cot^2 x - n + 1 = \frac{\cos^2 x}{\sin^2 x} - n + 1\)

Rearranging the Expression

Now, substituting this back into our integral, we have:

\[ \int \frac{\frac{\cos^2 x}{\sin^2 x} - n + 1}{(\cos x)^n} \, dx \]

This can be split into two separate integrals:

\[ \int \frac{\cos^2 x}{\sin^2 x (\cos x)^n} \, dx - \int \frac{n - 1}{(\cos x)^n} \, dx \]

Integrating Each Part

Let’s tackle these integrals one at a time. The first integral can be simplified further:

\[ \int \frac{\cos^2 x}{\sin^2 x (\cos x)^n} \, dx = \int \frac{1}{\sin^2 x} \cdot \frac{1}{(\cos x)^{n-2}} \, dx \]

Recognizing that \(\frac{1}{\sin^2 x}\) is the derivative of \(-\cot x\), we can use substitution or integration by parts if necessary. The second integral, \(-\int \frac{n - 1}{(\cos x)^n} \, dx\), can be approached using the integral of secant functions:

\[ -\int (n - 1) \sec^n x \, dx \]

Final Steps

For the first integral, we can use the identity \(\sec^2 x = 1 + \tan^2 x\) to help with integration. For the second integral, the integral of \(\sec^n x\) can be computed using reduction formulas or known results for specific values of \(n\).

In summary, the integration process involves breaking down the expression into simpler parts, applying trigonometric identities, and using known integration techniques for trigonometric functions. The final result will depend on the specific value of \(n\) and may require additional steps based on that value.

Example for Clarity

For instance, if \(n = 2\), the integral simplifies significantly:

\[ \int \frac{\cot^2 x - 1}{\cos^2 x} \, dx = \int \frac{\cot^2 x}{\cos^2 x} \, dx - \int \sec^2 x \, dx \]

Each part can be integrated separately, leading to a clearer solution. This approach can be adapted based on the value of \(n\) and the specific requirements of the problem.