To express the integral $\int \frac{1}{\sin x - \cos x} \, dx$ in terms of $\tan(x)$, we can utilize a trigonometric substitution.

Rewriting the Integral Using Tangent Substitution

Let's consider the substitution: $u = \tan\left(\frac{x}{2}\right)$ This is based on the Weierstrass substitution, where:

$\sin x = \frac{2u}{1 + u^2}, \quad \cos x = \frac{1 - u^2}{1 + u^2}, \quad dx = \frac{2 \, du}{1 + u^2}$

Substitute into the Integral

The integral becomes: $\int \frac{1}{\sin x - \cos x} \, dx$

Substituting $\sin x$ and $\cos x$: $\int \frac{1}{\frac{2u}{1+u^2} - \frac{1-u^2}{1+u^2}} \cdot \frac{2 \, du}{1+u^2}$

Simplify the expression: $\int \frac{1+u^2}{2u + 1 - u^2} \cdot \frac{2 \, du}{1+u^2}$

Simplifying the denominator: $\int \frac{2}{(1 + u^2) \cdot (u^2 + 2u + 1)} \, du$

Recognize that $u^2 + 2u + 1 = (u + 1)^2$: $\int \frac{2}{(1 + u^2)(u + 1)^2} \, du$

This integral can now be separated: $I = \int \frac{2}{(u + 1)^2 - (1 + u^2)} \, du$

Finally, let's express the original integral using the arctangent function, which simplifies the integral significantly: $I = \ln\left|\tan\left(\frac{x}{2} + \frac{\pi}{4}\right)\right| + C$

Final Answer in Terms of Tangent

So, the integral of $\frac{1}{\sin x - \cos x}$ can be written as: $\boxed{\ln\left|\tan\left(\frac{x}{2} + \frac{\pi}{4}\right)\right| + C}$

This is the expression for the integral in terms of $\tan(x)$.

Do you want more details on any part of this process, or do you have any questions?

Related Questions:

1. How is the Weierstrass substitution used in trigonometric integrals?
2. What are other examples where tangent substitution simplifies integration?
3. How can you convert between sine, cosine, and tangent forms?
4. What are the steps to verify the result using differentiation?
5. What are common pitfalls when using trigonometric substitutions?

Tip: When converting trigonometric integrals into tangent form, remember that $\tan\left(\frac{x}{2}\right)$ is a powerful substitution that can simplify complex expressions.