1
JEE Main 2021 (Online) 17th March Evening Shift
Numerical
+4
-1
Let $${I_n} = \int_1^e {{x^{19}}{{(\log |x|)}^n}} dx$$, where n$$\in$$N. If (20)I10 = $$\alpha$$I9 + $$\beta$$I8, for natural numbers $$\alpha$$ and $$\beta$$, then $$\alpha$$ $$-$$ $$\beta$$ equals to ___________.
2
JEE Main 2021 (Online) 17th March Morning Shift
Numerical
+4
-1
If [ . ] represents the greatest integer function, then the value of

$$\left| {\int\limits_0^{\sqrt {{\pi \over 2}} } {\left[ {[{x^2}] - \cos x} \right]dx} } \right|$$ is ____________.
3
JEE Main 2021 (Online) 16th March Morning Shift
Numerical
+4
-1
Let f : R $$\to$$ R be a continuous function such that f(x) + f(x + 1) = 2, for all x$$\in$$R.

If $${I_1} = \int\limits_0^8 {f(x)dx}$$ and $${I_2} = \int\limits_{ - 1}^3 {f(x)dx}$$, then the value of I1 + 2I2 is equal to ____________.
4
JEE Main 2021 (Online) 16th March Morning Shift
Numerical
+4
-1
Out of Syllabus
Let f : (0, 2) $$\to$$ R be defined as f(x) = log2$$\left( {1 + \tan \left( {{{\pi x} \over 4}} \right)} \right)$$. Then, $$\mathop {\lim }\limits_{n \to \infty } {2 \over n}\left( {f\left( {{1 \over n}} \right) + f\left( {{2 \over n}} \right) + ... + f(1)} \right)$$ is equal to ___________.