1
JEE Main 2022 (Online) 24th June Morning Shift
Numerical
+4
-1

Let $$f(\theta ) = \sin \theta + \int\limits_{ - \pi /2}^{\pi /2} {(\sin \theta + t\cos \theta )f(t)dt}$$. Then the value of $$\left| {\int_0^{\pi /2} {f(\theta )d\theta } } \right|$$ is _____________.

2
JEE Main 2022 (Online) 24th June Morning Shift
Numerical
+4
-1

Let $$\mathop {Max}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \alpha$$ and $$\mathop {Min}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \beta$$.

If $$\int\limits_{\beta - {8 \over 3}}^{2\alpha - 1} {Max\left\{ {{{9 - {x^2}} \over {5 - x}},x} \right\}dx = {\alpha _1} + {\alpha _2}{{\log }_e}\left( {{8 \over {15}}} \right)}$$ then $${\alpha _1} + {\alpha _2}$$ is equal to _____________.

3
JEE Main 2021 (Online) 31st August Morning Shift
Numerical
+4
-1
Let [t] denote the greatest integer $$\le$$ t. Then the value of

$$8.\int\limits_{ - {1 \over 2}}^1 {([2x] + |x|)dx}$$ is ___________.
4
JEE Main 2021 (Online) 31st August Morning Shift
Numerical
+4
-1
If $$x\phi (x) = \int\limits_5^x {(3{t^2} - 2\phi '(t))dt}$$, x > $$-$$2, and $$\phi$$(0) = 4, then $$\phi$$(2) is __________.