1
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 _____________.
2
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
3
JEE Main 2021 (Online) 16th March Evening Shift
Numerical
+4
-1
For real numbers $$\alpha$$, $$\beta$$, $$\gamma$$ and $$\delta$$, if
$$\int {{{({x^2} - 1) + {{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)} \over {({x^4} + 3{x^2} + 1){{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)}}dx}$$

$$= \alpha {\log _e}\left( {{{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)} \right) + \beta {\tan ^{ - 1}}\left( {{{\gamma ({x^2} + 1)} \over x}} \right) + \delta {\tan ^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right) + C$$

where C is an arbitrary constant, then the value of 10($$\alpha$$ + $$\beta$$$$\gamma$$ + $$\delta$$) is equal to ______________.
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