Integral Calculation - Exercise 20

We are asked to evaluate the following indefinite integral:

$$ \int \frac{\sin x + \cos x}{\sin x - \cos x} \ dx $$

We’ll use the differential method, focusing on the expression $ \sin x - \cos x $ as a substitution candidate.

First, compute its differential:

$$ d( \sin x - \cos x ) = ( \cos x + \sin x ) \ dx $$

Now, solve for dx by dividing both sides by $ \cos x + \sin x $:

$$ \frac{ d( \sin x - \cos x ) }{ \cos x + \sin x } = dx $$

We substitute this expression for dx back into the integral:

$$ \int \frac{\sin x + \cos x}{\sin x - \cos x} \cdot \left[ \frac{ d( \sin x - \cos x ) }{ \cos x + \sin x } \right] $$

This allows us to cancel the numerator and denominator:

$$ \int \frac{1}{\sin x - \cos x} \cdot d( \sin x - \cos x ) $$

Next, let’s set a new variable: $ t = \sin(x) - \cos(x) $.

$$ \int \frac{1}{t} \ dt $$

This is a standard form, whose antiderivative is:

$$ \int \frac{1}{t} \ dt = \log |t| + c $$

Substituting back $ t = \sin(x) - \cos(x) $, we obtain:

$$ \log | \sin x - \cos x | + c $$

So the final solution is:

$$ \int \frac{\sin x + \cos x}{\sin x - \cos x} \ dx = \log | \sin x - \cos x | + c $$

Here’s a quicker method, suggested by a user.

$$ \int \frac{\sin x + \cos x}{\sin x - \cos x} \ dx $$

Notice that the numerator $ \sin(x) + \cos(x) $ is exactly the derivative of the denominator $ \sin(x) - \cos(x) $.

This allows us to apply one of the standard integration rules directly:

$$ \int \frac{f'(x)}{f(x)} \ dx = \log |f(x)| + c $$

(Here, "log" refers to the natural logarithm, i.e., ln.)

This means the integrand is the derivative of the composite function $ \log[\sin(x) - \cos(x)] $.

To confirm this, differentiate explicitly:

$$ \frac{d}{dx} \log[\sin(x) - \cos(x)] = \frac{1}{\sin(x) - \cos(x)} \cdot \frac{d}{dx}[\sin(x) - \cos(x)] $$

$$ = \frac{1}{\sin(x) - \cos(x)} \cdot [\cos(x) + \sin(x)] $$

$$ = \frac{\sin(x) + \cos(x)}{\sin(x) - \cos(x)} $$

Therefore, we once again conclude:

$$ \int \frac{\sin x + \cos x}{\sin x - \cos x} \ dx = \log | \sin x - \cos x | + c $$

And so on...