Question

integration question, equation attached

Answer 1

there is no equation attached

Answer 2

Let n be an arbitrary integer. If the substitution u = 1 + x^(1/n) is made in the integral

Answer 3

more coming...this thing is acting strange

Answer 4

...continued...\[\int\limits_{}^{}\sqrt{1 + x ^{1/n}}dx\] is made, what is the new integral?

Answer 5

that's the entire question

Answer 6

do you need to solve that integral?

Answer 7

just need to find the new integral

Answer 8

so, is it just \[\int\limits_{}^{}\sqrt{u}\]

Answer 9

du = (n/(n + 1))x^((n + 1)/n) dx

Answer 10

could I solve for dx, then substitute that, and multiply it by sqrt(u)?

Answer 11

we have the subsequent steps:
\[\Large \begin{gathered}
u = 1 + {x^{1/n}} \hfill \\
\hfill \\
du = \frac{1}{n}{x^{\frac{1}{n} - 1}}dx = \frac{1}{n}\frac{{{x^{\frac{1}{n}}}}}{x}dx = \frac{1}{n}\frac{{u - 1}}{{{{\left( {u - 1} \right)}^n}}}dx \hfill \\
\hfill \\
dx = n{\left( {u - 1} \right)^{n - 1}}du \hfill \\
\end{gathered} \]

Answer 12

so your integral becames:
\[\Large \int {n\sqrt u } {\left( {u - 1} \right)^{n - 1}}du\]

Answer 13

I understand, thank you for the help

Answer 14

thank you!