Mathematics

# If $P=\displaystyle \lim _{ n\rightarrow \infty }{ \frac { { \left( \prod _{ r=1 }^{ n }{ \left( { n }^{ 3 }+{ r }^{ 3 } \right) } \right) }^{ 1/n } }{ { n }^{ 3 } } }$  and $\lambda =\displaystyle \int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 3 } } }$ then $In P$ is equal to

$In 2-1+\lambda$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Subjective Medium
Evaluate the following integral:
$\displaystyle \int { \cfrac { \sin ^{ 2 }{ x } }{ 1+\cos { x } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int_0^a \dfrac{x^4}{a^2+x^2)^4}dx$
• A. $\dfrac{1}{4a^3}\left(\dfrac{\pi}{4}+\dfrac{1}{3}\right)$
• B. $\dfrac{1}{16a^3}\left(\dfrac{\pi}{4}-\dfrac{1}{3}\right)$
• C. $\dfrac{1}{4a^3}\left(\dfrac{\pi}{4}-\dfrac{1}{3}\right)$
• D. $\dfrac{1}{16a^3}\left(\dfrac{\pi}{4}+\dfrac{1}{3}\right)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\displaystyle\int^{\pi/2}_0(2log\sin x-log \sin 2x)dx$ equals.
• A. $\pi log 2$
• B. $-\pi log 2$
• C. $(\pi/2)log 2$
• D. $-(\pi/2)log 2$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
$\int {x{{\sin }^{ - 1}}xdx}$

Find $\int { x\sqrt { 1+x-{ x }^{ 2 } } dx }$