1
JEE Main 2023 (Online) 12th April Morning Shift
Numerical
+4
-1 Let $$I(x)=\int \sqrt{\frac{x+7}{x}} \mathrm{~d} x$$ and $$I(9)=12+7 \log _{e} 7$$. If $$I(1)=\alpha+7 \log _{e}(1+2 \sqrt{2})$$, then $$\alpha^{4}$$ is equal to _________.

2
JEE Main 2023 (Online) 30th January Evening Shift
Numerical
+4
-1 If $\int \sqrt{\sec 2 x-1} d x=\alpha \log _e\left|\cos 2 x+\beta+\sqrt{\cos 2 x\left(1+\cos \frac{1}{\beta} x\right)}\right|+$ constant, then $\beta-\alpha$ is equal to ____________.
3
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 ______________.
4
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 _____________.
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