JEE Main 2023 (Online) 1st February Morning Shift | Definite Integration Question 61 | Mathematics

$$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} \right]$$ is equal to

$${\log _e}2$$

$${\log _e}\left( {{2 \over 3}} \right)$$

$${\log _e}\left( {{3 \over 2}} \right)$$

JEE Main 2023 (Online) 31st January Evening Shift

MCQ (Single Correct Answer)

-1

Let $\alpha>0$. If $\int\limits_0^\alpha \frac{x}{\sqrt{x+\alpha}-\sqrt{x}} \mathrm{~d} x=\frac{16+20 \sqrt{2}}{15}$, then $\alpha$ is equal to :

$2 \sqrt{2}$

$\sqrt{2}$

JEE Main 2023 (Online) 31st January Evening Shift

MCQ (Single Correct Answer)

-1

If $\phi(x)=\frac{1}{\sqrt{x}} \int\limits_{\frac{\pi}{4}}^x\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t)\right) d t, x>0$,

then $\emptyset^{\prime}\left(\frac{\pi}{4}\right)$ is equal to :

$\frac{4}{6+\sqrt{\pi}}$

$\frac{4}{6-\sqrt{\pi}}$

$\frac{8}{\sqrt{\pi}}$

$\frac{8}{6+\sqrt{\pi}}$

JEE Main 2023 (Online) 31st January Morning Shift

MCQ (Single Correct Answer)

-1

Let $$\alpha \in (0,1)$$ and $$\beta = {\log _e}(1 - \alpha )$$. Let $${P_n}(x) = x + {{{x^2}} \over 2} + {{{x^3}} \over 3}\, + \,...\, + \,{{{x^n}} \over n},x \in (0,1)$$. Then the integral $$\int\limits_0^\alpha {{{{t^{50}}} \over {1 - t}}dt} $$ is equal to

$$ - \left( {\beta + {P_{50}}\left( \alpha \right)} \right)$$

$$\beta - {P_{50}}(\alpha )$$

$${P_{50}}(\alpha ) - \beta $$

$$\beta + {P_{50}} - (\alpha )$$

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