1
JEE Main 2022 (Online) 25th July Evening Shift
Numerical
+4
-1

Let $${a_n} = \int\limits_{ - 1}^n {\left( {1 + {x \over 2} + {{{x^2}} \over 3} + \,\,.....\,\, + \,\,{{{x^{n - 1}}} \over n}} \right)dx}$$ for every n $$\in$$ N. Then the sum of all the elements of the set {n $$\in$$ N : an $$\in$$ (2, 30)} is ____________.

2
JEE Main 2022 (Online) 25th July Morning Shift
Numerical
+4
-1
Out of Syllabus

\begin{aligned} &\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)] \\ &=33 \cdot \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot\left[1^{k}+2^{k}+3^{k}+\ldots+n^{k}\right] \end{aligned}, then the integral value of $$\mathrm{k}$$ is equal to _____________

3
JEE Main 2022 (Online) 30th June Morning Shift
Numerical
+4
-1

Let $$f(t) = \int\limits_0^t {{e^{{x^3}}}\left( {{{{x^8}} \over {{{({x^6} + 2{x^3} + 2)}^2}}}} \right)dx}$$. If $$f(1) + f'(1) = \alpha e - {1 \over 6}$$, then the value of 150$$\alpha$$ is equal to ___________.

4
JEE Main 2022 (Online) 26th June Evening Shift
Numerical
+4
-1

The integral $${{24} \over \pi }\int_0^{\sqrt 2 } {{{(2 - {x^2})dx} \over {(2 + {x^2})\sqrt {4 + {x^4}} }}}$$ is equal to ____________.