1
JEE Main 2021 (Online) 31st August Evening Shift
Numerical
+4
-1
If $$\int {{{\sin x} \over {{{\sin }^3}x + {{\cos }^3}x}}dx = }$$

$$\alpha {\log _e}|1 + \tan x| + \beta {\log _e}|1 - \tan x + {\tan ^2}x| + \gamma {\tan ^{ - 1}}\left( {{{2\tan x - 1} \over {\sqrt 3 }}} \right) + C$$, when C is constant of integration, then the value of $$18(\alpha + \beta + {\gamma ^2})$$ is ______________.
2
JEE Main 2021 (Online) 27th August Evening Shift
Numerical
+4
-1
If $$\int {{{2{e^x} + 3{e^{ - x}}} \over {4{e^x} + 7{e^{ - x}}}}dx = {1 \over {14}}(ux + v{{\log }_e}(4{e^x} + 7{e^{ - x}})) + C}$$, where C is a constant of integration, then u + v is equal to _____________.
3
JEE Main 2021 (Online) 27th August Morning Shift
Numerical
+4
-1
If $$\int {{{dx} \over {{{({x^2} + x + 1)}^2}}} = a{{\tan }^{ - 1}}\left( {{{2x + 1} \over {\sqrt 3 }}} \right) + b\left( {{{2x + 1} \over {{x^2} + x + 1}}} \right) + C}$$, x > 0 where C is the constant of integration, then the value of $$9\left( {\sqrt 3 a + b} \right)$$ is equal to _____________.
4
JEE Main 2021 (Online) 18th March Morning Shift
Numerical
+4
-1
If $$f(x) = \int {{{5{x^8} + 7{x^6}} \over {{{({x^2} + 1 + 2{x^7})}^2}}}dx,(x \ge 0),f(0) = 0}$$ and $$f(1) = {1 \over K}$$, then the value of K is