Question

Please, help me with that =)
Integrate dx/(a+bx^n)^((n+1)/n) with a different from 0}
Thank you guys

Answer 1

\(\displaystyle\int\frac{dx}{(a+bx^n)^{(n+1)/n}}\) with \(a\neq0\) ?

Answer 2

*

Answer 3

In the denominator, take
\[\left(a+bx^n\right)^{(n+1)/n}=\left(x^n \right)^{(n+1)/n}\left(\frac{a}{x^n}+b\right)^{(n+1)/n}=x^{n+1}\left(\frac{a}{x^n}+b\right)^{(n+1)/n}\]
Then substitute \(u=\dfrac{a}{x^n}+b\).

Answer 4

\[\begin{align*}\int\frac{dx}{\left(a+bx^n\right)^{(n+1)/n}}&=\int\frac{dx}{x^{n+1}\left(\dfrac{a}{x^n}+b\right)^{(n+1)/n}}\\\\
&=-\frac{1}{an}\int\frac{du}{u^{(n+1)/n}}
\end{align*}\]

Answer 5

Notice that \(\dfrac{n+1}{n}\neq1\) for any \(n\), so you can immediately use the power rule and not have to worry about the antiderivative form of \(\dfrac{1}{u}\).

Answer 6

Thank you very much for your help.