Find the limit of
limn→∞∫10(1+xn)ndx
Let u= 1 +\frac{x}{n} \implies du =\frac{1}{n} dx \implies n \cdot du = dx
at x=0 u=1 and at x=1 u=1+\frac{1}{n} so now limit will change from 1 to 1+\frac{1}{n}
Back to the integral
\lim\limits_{n \rightarrow \infty} \left( n \cdot \int_{1}^{1+\frac{1}{n}} u^n du \right)= \lim\limits_{n \rightarrow \infty} \left( n \cdot \left[ \frac{nu^{n+1}}{n+1} \right]_1^{1+\frac{1}{n}} \right) = \lim\limits_{n \rightarrow \infty} \left(\frac{n^2}{n+1} \left[ u^{n+1} \right]_1^{1+\frac{1}{n}} \right)
\implies\lim\limits_{n \rightarrow \infty} \left(\frac{n^2}{n+1} \left[ \left(1+\frac{1}{n} \right)^{n+1}-1 \right] \right)=\infty
Is my finding correct? Is the procedure of taking the limit before completing the integration correct?
Much appreciated
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