let $\alpha'(x)=\beta(x), \beta'(x)=\alpha(x)$ and assume that $\alpha^2 - \beta^2 = 1$. how would I go about calculating the following anti derivative : $\int (\alpha (x))^5 (\beta(x))^4$d$x$.
I have tried many different methods with no success. Firstly, I tried to write the integral in terms of one of the variables (using the first condition) and then using integration by parts. This resulted in a very complex answer which I abandoned midway. Additionally, I have tried to use an appropriate U-substitution which unfortunately does not seem to work. This is because when I take the first derivative of my u - due to chain rule - I get a product with one variable being either $\alpha'(x)$ or $\beta'(x)$ and as a result I cannot continue further with the integration due to not being able to get the entire integral in terms of only u.
Any help would be appreciated.
Thank you.
$$ \int (\alpha (x))^5(\beta (x))^4dx= \int \alpha (x)(1+(\beta (x))^2)^2(\beta (x))^4dx $$ Whith the substitution $u=\beta(x)$, so $du=\beta'(x)dx=\alpha(x)dx$ you have:$$ \int (1+u^2)^2u^4du=\frac{1}{5}u^5+\frac{2}{7}u^7+\frac{1}{9}u^9+C $$ Finally: $$ \int (\alpha (x))^5(\beta (x))^4dx=\frac{1}{5}(\beta (x))^5+\frac{2}{7}(\beta (x))^7+\frac{1}{9}(\beta (x))^9+C $$ If you want you can also express everything in terms of $\alpha$, but it would add nothing to the solution.