1
JEE Main 2023 (Online) 31st January Evening Shift
+4
-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 :
A
$\frac{4}{6+\sqrt{\pi}}$
B
$\frac{4}{6-\sqrt{\pi}}$
C
$\frac{8}{\sqrt{\pi}}$
D
$\frac{8}{6+\sqrt{\pi}}$
2
JEE Main 2023 (Online) 31st January Morning Shift
+4
-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

A
$$- \left( {\beta + {P_{50}}\left( \alpha \right)} \right)$$
B
$$\beta - {P_{50}}(\alpha )$$
C
$${P_{50}}(\alpha ) - \beta$$
D
$$\beta + {P_{50}} - (\alpha )$$
3
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1

The value of $$\int_\limits{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x$$ is equal to :

A
$$\frac{10}{3}-\sqrt{3}+\log _{e} \sqrt{3}$$
B
$$\frac{7}{2}-\sqrt{3}-\log _{e} \sqrt{3}$$
C
$$\frac{10}{3}-\sqrt{3}-\log _{e} \sqrt{3}$$
D
$$-2+3\sqrt{3}+\log _{e} \sqrt{3}$$
4
JEE Main 2023 (Online) 30th January Evening Shift
+4
-1
Out of Syllabus
$\lim\limits_{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}$ is equal to :
A
0
B
$\frac{19}{3}$
C
19
D
12
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