1
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The value of the integral $$\int_\limits{-\log _{e} 2}^{\log _{e} 2} e^{x}\left(\log _{e}\left(e^{x}+\sqrt{1+e^{2 x}}\right)\right) d x$$ is equal to :

A
$$\log _{e}\left(\frac{(2+\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$$
B
$$\log _{e}\left(\frac{\sqrt{2}(2+\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$$
C
$$\log _{e}\left(\frac{2(2+\sqrt{5})}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$$
D
$$\log _{e}\left(\frac{\sqrt{2}(3-\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$$
2
JEE Main 2023 (Online) 10th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$f$$ be a continuous function satisfying $$\int_\limits{0}^{t^{2}}\left(f(x)+x^{2}\right) d x=\frac{4}{3} t^{3}, \forall t > 0$$. Then $$f\left(\frac{\pi^{2}}{4}\right)$$ is equal to :

A
$$-\pi\left(1+\frac{\pi^{3}}{16}\right)$$
B
$$\pi\left(1-\frac{\pi^{3}}{16}\right)$$
C
$$-\pi^{2}\left(1+\frac{\pi^{2}}{16}\right)$$
D
$$\pi^{2}\left(1-\frac{\pi^{2}}{16}\right)$$
3
JEE Main 2023 (Online) 6th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$f(x)$$ be a function satisfying $$f(x)+f(\pi-x)=\pi^{2}, \forall x \in \mathbb{R}$$. Then $$\int_\limits{0}^{\pi} f(x) \sin x d x$$ is equal to :

A
$$\pi^{2}$$
B
$$\frac{\pi^{2}}{2}$$
C
$$2 \pi^{2}$$
D
$$\frac{\pi^{2}}{4}$$
4
JEE Main 2023 (Online) 6th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

$$\lim _\limits{n \rightarrow \infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right) \ldots . .\left(2^{\frac{1}{2}}-2^{\frac{1}{2 n+1}}\right)\right\}$$ is equal to :

A
$$\sqrt{2}$$
B
1
C
$$\frac{1}{\sqrt{2}}$$
D
0
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