Integration by substitution: problem

Question

Let $a>1$:

$\int_{x}^{ax} \frac{ax-t}{t^2}dt=\int_{1}^{a} \frac{a-t}{t^2}dt$.

I have as a hint to substitute with $t=s*x$, but by substituting I don't actually get $dt$ on the right side then?

Am a little bit confused.

Answer 1

For the left hand side, substitute $t$ by $sx$. $$\begin{align} LHS&=\int_{1}^{a} \frac{ax-sx}{(sx)^2} x \operatorname{d}s\\ &=\int_{1}^{a} \frac{a-s}{s^2} \operatorname{d}s \end{align}$$

Actually the outcome is the same as the right hand side. The only thing you need to do is simply replace "t" with "s" in the right hand side, since it will not affect the final result.

$$\begin{align} \int_{1}^{a} \frac{a-s}{s^2} \operatorname{d}s=\int_{1}^{a} \frac{a-t}{t^2} \operatorname{d}t&=\int_{1}^{a} \frac{a}{t^2} -\frac{1}{t}\operatorname{d}t\\ &=(-\frac{a}{t}-\log t)\bigg|_{t=1}^{a}\\ &=a-1-\log a \end{align}$$