Integral of sin x cos x / (sin x - cos x^n) dx: An Exploration of Integration Techniques

In the realm of calculus, evaluating integrals can be both a challenge and a rewarding endeavor. One interesting integral is (int frac{sin x cos x}{sin x - cos^n x} dx). This article explores this integral in detail, breaking it down into manageable steps and employing various integration techniques.

Introduction to the Integral

The integral in question, (int frac{sin x cos x}{sin x - cos^n x} dx), is a classic example in advanced calculus. It involves trigonometric functions, making it particularly intriguing. Let's delve into the steps required to solve this integral.

Step 1: Substitution

To simplify the integral, we will use a substitution. Let's set (t sin x - cos^n x). The goal of this substitution is to simplify the denominator of the given integral.

Deriving dt

First, we need to find the differential (dt) in terms of (dx).

dt  d(sin x - cos^n x)

Using the chain rule, we get:

dt  cos x dx - n cos^{n-1} x cdot (-sin x) dx

Simplifying, we obtain:

dt  (cos x   n cos^{n-1} x sin x) dx

Note that (cos x sin x dx) is a part of (dt), which we will leverage later.

Step 2: Rewriting the Integral

With (t sin x - cos^n x), we need to adjust our integral accordingly.

The original integral is:

(int frac{sin x cos x}{sin x - cos^n x} dx  int frac{sin x cos x}{t} dx)

Now, using the derived (dt), we can substitute and simplify:

(int frac{sin x cos x}{t} dx)

Since (cos x sin x dx dt - n cos^{n-1} x sin x dx), we can simplify the integral to:

(int frac{1}{t} dt)

Step 3: Integration

The integral now simplifies to a standard form:

(int frac{1}{t} dt  ln |t|   C)

Substituting back (t sin x - cos^n x), we get:

(ln |sin x - cos^n x|   C)

Conclusion

In conclusion, the solution to the integral (int frac{sin x cos x}{sin x - cos^n x} dx) is:

(ln |sin x - cos^n x|   C)

This solution leverages the substitution method to transform the integral into a more manageable form, allowing us to integrate it effectively. The steps provided give a clear pathway to solving this type of integral, which is a valuable skill in advanced calculus and mathematical problem-solving.

Keywords

integration, trigonometric functions, calculus techniques