# Thread: Integration by substitution

1. ## Integration by substitution

Hi

I have this expression but don't know how the answer was reached.

$\displaystyle \int\frac{e^{a/x}}{x^{2}}=-\frac{e^{a/x}}{a}$

I'm sure substitution is involved, this is a weak area for me.

Any help would be greatly appreciated.

Thanks

2. The derivative of $\displaystyle e^{g(x)}$ is $\displaystyle e^{g(x)}\cdot g^{'}(x)$... all right?... and which is the derivative of $\displaystyle \frac{a}{x}$?...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. Use the substitution $\displaystyle \frac{a}{x}=t$.

$\displaystyle \frac{1}{x}=\frac{t}{a}\Rightarrow-\frac{1}{x^2}dx=\frac{1}{a}dt\Rightarrow \frac{1}{x^2}dx=-\frac{1}{a}dt$.

Then $\displaystyle \int e^t\left(-\frac{1}{a}\right)dt=-\frac{1}{a}\int e^tdt=-\frac{e^t}{a}+C$

Now replace back t with $\displaystyle \frac{a}{x}$.