Example 1:
$$\int\frac{\tan^2(\ln x)}{x}dx$$
Method 1:
$$\text{Let}\ u=\ln x$$
$$\implies du=d(\ln x)$$
$$\implies du=\ln'(x)dx$$
$$\implies du=\frac{1}{x}dx$$
Method 2:
$$\text{Let}\ u=\ln x\tag{Start}$$
$$\implies \frac{du}{dx}=\frac{1}{x}$$
$$\implies xdu=dx$$
$$\implies du=\frac{dx}{x}\tag{End}$$
Example 2
$$\int\frac{dx}{x\sqrt{x^4-1}}$$
Method 1:
$$\text{Let}\ x^4=\frac{1}{u^2}$$
$$\implies u=\frac{1}{x^2}$$
$$\implies du=d(\frac{1}{x^2})$$
$$\implies du=d(x^{-2})$$
$$\implies du=-2x^{-3}dx$$
Method 2:
$$\text{Let}\ x^4=\frac{1}{u^2}\tag{Start}$$
$$\implies 4x^3=-2u^{-3}\frac{du}{dx}$$
$$\implies 2x^3=-u^{-3}\frac{du}{dx}$$
$$\implies 2x^3dx=-u^{-3}du$$
$$\implies du=-\frac{2x^3dx}{u^{-3}}$$
$$\implies du=-\frac{2x^3dx}{(\frac{1}{x^2})^{-3}}$$
$$\implies du=-2x^{-3}dx\tag{End}$$
Questions:
- In both examples 1 and 2, in method 1, I just used this the rule $d(f(t)) = f'(t)dt$. However, in method 2, I used differentiation in both sides, cross multiplication, dividing both sides by $x$, other general rules we use in equations etc. I'm certain that method 1 is correct. In both method 1 and 2, I got the same end expression, so the result of method 2 is also correct. However, are the intermediate lines between the start and the end correct in method 2? I'm asking this because I used cross multiplication and other techniques; I managed to arrive at the correct answer, but I'm anxious whether the intermediate lines are correct to write or not i.e. is LHS=RHS true in those lines.
The reason for why you want to find a $\mathrm{d}u$ from a $\mathrm{d}x$ is in reality because you have the substitution theorem which tells you that, simply put,
$$\int f(u(x))\frac{\mathrm{d}u(x)}{\mathrm{d}x}\mathrm{d}x=\int f(u)\mathrm{d}u.$$
Thus what you do in reality when you do a substitution, i.e. there is no actual fraction which you can cancel things in. However this theorem tells you that you can essentially treat it like a fraction when you do a substitution, but formally it is your second method which is more along the lines of what is actually going on.