calculus - When am I allowed to use ln(x) when integrating functions?

In Mathematics, we know the following is true:

$$\int \frac{1}{x} \space dx = \ln(x)$$

Not only that, this rule works for constants added to x:
$$\int \frac{1}{x + 1}\space dx = \ln(x + 1) + C{3}$$
$$\int \frac{1}{x + 3}\space dx = \ln(x + 3) + C$$
$$\int \frac{1}{x - 37}\space dx = \ln(x - 37) + C$$
$$\int \frac{1}{x - 42}\space dx = \ln(x - 42) + C$$

So its pretty safe to say that $$\int \frac{1}{x + a}\space dx = \ln(x + a) + C$$ But the moment I introduce $x^a$ where $a$ is not equal to 1, the model collapses. The integral of $1/x^a$ is not equal to $\ln(x^a)$. The same goes for $\cos(x)$, and $\sin(x)$, and other trig functions.

So when are we allowed or not allowed to use the rule of $\ln(x)$ when integrating functions?

Answer

Generally speaking, "using $\ln (x)$" as a rule or technique is unheard of. When one speaks of techniques, they usually include integration by substitution, integration by parts, trig substitutions, partial fractions, etc. With introductory calculus in mind, $\ln |x|$ is defined as $\int \frac{1}{x} \ dx.$ This can be extended to $\ln |u| = \int \frac{1}{u} \ du.$ Note that there are many more definitions for $\ln (x)$, but I felt this best related particularly to your examples.

For your first couple of examples, when choosing your $u$ to be the denominator, the $du$ is simply equal to $dx.$ This is what 'allows' the integrand to be evaluated to just $\ln |u|$ where $u$ is a linear expression.

In regards to $\int \frac{1}{x^2 + a} \ dx$, this can be handled using an inverse tangent and would be evaluated to
$$\dfrac{\operatorname{arctan}(\frac{x}{\sqrt{a}})}{\sqrt{a}} + C$$

For integrals of the form
$$\int \frac{1}{x^n + a} \ dx$$

where $n \ge 3$, you will have to revert to partial fractions. For more on partial fractions, see this.