# I am confused on the substitution method for the following integral. I've put it in symbolab and wolframalpha and idk how they get -1/2*e^(2/x^2)

Here is the integral:

$$\int \frac{2e^{\frac{2}{x^2}}}{x^3}dx$$ or integral of $$[2e^{(2/x^2)}]/x^3\,dx$$

When I tried solving for it with substitution, I simply used $$u=2/x^2$$ whilst symbolab seemingly made $$u=x$$ and made $$v=2/x^2$$ and turned the equation into $$2\;*$$ integral of $$[e^{(2/u)}]/2u^2\,du$$ and then further manipulated the equation into $$2\,* [1/2]$$ integral of $$- [e^v]/2\,dv$$. From that step, I sort of understood how the answer $$-1/2*e^{(2/x^2)}$$ was found but I don't really understand how "$$u$$" went to "$$v$$" in terms of substitution. How would you solve the equation solely using "$$u$$" substitution? (For context: I am a high-school senior taking a brief calculus course and this is from a test that was assessing "$$u$$" substitution and I realized the $$u/du$$ pattern did not apply to this problem).

• May 2, 2023 at 18:49

Using your substitution $$u=\frac{2}{x^2}$$ you should get $$du=-\frac{4 \ dx}{x^3}$$, $$-\frac{1}{4}du=\frac{dx}{x^3}$$. Also, $$2e^{\frac{2}{x^2}}=2e^u$$ and putting it all together, the integral becomes $$-\frac{1}{2}\int e^udu$$