1
JEE Main 2023 (Online) 8th April Evening Shift
Numerical
+4
-1

Let $$[t]$$ denote the greatest integer function. If $$\int_\limits{0}^{2.4}\left[x^{2}\right] d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}$$, then $$\alpha+\beta+\gamma+\delta$$ is equal to __________.

2
JEE Main 2023 (Online) 8th April Morning Shift
Numerical
+4
-1

Let $$[t]$$ denote the greatest integer $$\leq t$$. Then $$\frac{2}{\pi} \int_\limits{\pi / 6}^{5 \pi / 6}(8[\operatorname{cosec} x]-5[\cot x]) d x$$ is equal to __________.

3
JEE Main 2023 (Online) 6th April Evening Shift
Numerical
+4
-1
Out of Syllabus

Let $$f(x)=\frac{x}{\left(1+x^{n}\right)^{\frac{1}{n}}}, x \in \mathbb{R}-\{-1\}, n \in \mathbb{N}, n > 2$$.

If $$f^{n}(x)=\left(f \circ f \circ f \ldots .\right.$$. upto $$n$$ times) $$(x)$$, then

$$\lim _\limits{n \rightarrow \infty} \int_\limits{0}^{1} x^{n-2}\left(f^{n}(x)\right) d x$$ is equal to ____________.

4
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1

If $$\int\limits_0^\pi {{{{5^{\cos x}}(1 + \cos x\cos 3x + {{\cos }^2}x + {{\cos }^3}x\cos 3x)dx} \over {1 + {5^{\cos x}}}} = {{k\pi } \over {16}}}$$, then k is equal to _____________.