derive the reduction formular for integral of( cos^m x/(sin^n x sin^m x)}dx

To derive the reduction formula for the integral 

\[
I_{m,n} = \int \frac{\cos^m x}{\sin^n x \sin^m x} \, dx
\]

let's break it down step by step. 

### Step 1: Simplify the Integrand
Notice that the integrand can be simplified as:

\[
I_{m,n} = \int \frac{\cos^m x}{\sin^{m+n} x} \, dx
\]

This can also be rewritten as:

\[
I_{m,n} = \int \cos^m x \sin^{-(m+n)} x \, dx
\]

### Step 2: Use Substitution
Let's use the substitution \( u = \sin x \), hence \( du = \cos x \, dx \). Then the integral becomes:

\[
I_{m,n} = \int u^{-(m+n)} \left(1 - u^2\right)^{\frac{m}{2}} du
\]

### Step 3: Apply Integration by Parts (Reduction Formula)
Let's try to use integration by parts to derive the reduction formula. Set:

\[
v = \sin^{-(m+n-1)} x \quad \text{and} \quad dw = \sin^{-(m+n)} x \cos^{m-1} x \, dx
\]

First, we apply the integration by parts formula:

\[
\int v \, dw = v \cdot w - \int w \, dv
\]

So:

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

### Step 4: Simplify the Expression
Thus, the reduction formula becomes:

\[
I_{m,n} = -\frac{\cos^{m-1} x}{(m+n-1)\sin^{m+n-1} x} + \frac{m-1}{m+n-1} I_{m-2,n}
\]

This is the reduction formula for the integral \( I_{m,n} \).

### Summary
The reduction formula for 

\[
I_{m,n} = \int \frac{\cos^m x}{\sin^n x \sin^m x} \, dx
\]

is given by:

\[
I_{m,n} = -\frac{\cos^{m-1} x}{(m+n-1)\sin^{m+n-1} x} + \frac{m-1}{m+n-1} I_{m-2,n}
\]

This formula allows you to express the integral in terms of a simpler integral with lower powers of \( m \).

---

### Follow-Up Questions:
1. How does the reduction formula change if \( m = n \)?
2. Can this formula be applied for \( m \) and \( n \) being odd or even? How does the result differ?
3. What specific cases can be directly computed using this reduction formula?
4. How would the reduction formula change if we had an extra factor, like \( \cos^p(x) \), in the denominator?
5. How can this reduction formula be used to compute the definite integral over the interval \( [0, \frac{\pi}{2}] \)?

### Tip:
When deriving reduction formulas, always check the boundary conditions and simplifications for special cases (like when \( m \) or \( n \) is small) to verify the formula's accuracy.

Learn how to derive the reduction formula for the integral of cosine powers over sine products. This detailed solution covers the step-by-step derivation process, including substitution and integration by parts. Suitable for advanced college-level students and mathematicians.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation