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  • Let <img alt="\lambda= \int_{0}^{1}\frac{dx}{1+x^{3}}, \; p=\lim_{x\rightarrow \infty }\left [ \frac{\prod_{r=1}^{n}(n^{3}+r^{3})}{n^{3n}} \right ]^{1/n}" src="https://learn.careers360.com/latex-im

Let \lambda= \int_{0}^{1}\frac{dx}{1+x^{3}}, \; p=\lim_{x\rightarrow \infty }\left [ \frac{\prod_{r=1}^{n}(n^{3}+r^{3})}{n^{3n}} \right ]^{1/n}, Then \ln p is equal to

Option: 1

\ln2-1+\lambda


Option: 2

\ln2-3+3\lambda


Option: 3

2\ln2-\lambda


Option: 4

2\ln4-3+3\lambda


Answers (1)

best_answer

 

Integration By PARTS -

Let u and v are two functions then 

\int u\cdot vdx=u\int vdx-\int \left ( \frac{du}{dx}\int vdx \right )dx

- wherein

Where u is the Ist function v is he IInd function

 

 

P=\lim_{n\rightarrow \infty }\left [ \frac{\prod_{r=1}^{n}(3^{3}+r^{3})}{3^{3n}} \right ]^{1/n}

\ln p=\lim_{n\rightarrow \infty }.\frac{1}{n}\sum_{r=1}^{n}\ln\left ( 1+\left ( \frac{r}{n} \right )^{3} \right )=\int_{0}^{1}\ln(1+x^{3})dx

= \ln 2-3+3\lambda

Posted by

Anam Khan

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