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The integral \( \dfrac{24}{\pi} \displaystyle \int_{0}^{\sqrt{2}} \dfrac{(2 - x^2)\, dx}{(2 + x^2)\sqrt{4 + x^4}} \) is equal to _______.

For \( I(x)=\displaystyle \int \frac{\sec^2 x-2022}{\sin^{2022}x}\,dx \), if \( I\!\left(\dfrac{\pi}{4}\right)=2^{1011} \), then

Let $g:(0,\infty) \rightarrow R$ be a differentiable function such that

$\int \left( \frac{x(\cos x - \sin x)}{e^x + 1} + \frac{g(x)(e^x + 1 - xe^x)}{(e^x + 1)^2} \right) dx = \frac{x \, g(x)}{e^x + 1} + c \text{ for all } x > 0 \text{, where } c \text{ is an arbitrary constant. Then :}$

If \( \int \dfrac{(x^2+1)e^x}{(x+1)^2}\,dx = f(x)e^x + C \), where \( C \) is a constant, then \( \dfrac{d^3 f}{dx^3} \) at \( x=1 \) is equal to:

If \( \int{}^{} \frac{1}{x}\sqrt{\frac{1-x}{1+x}} \, dx = g(x) + c, \, g(1) = 0 \), then \( g\!\left(\tfrac{1}{2}\right) \) is equal to:

Let \( f(x)=\min\{[x-1],[x-2],\ldots,[x-10]\} \) where \( [t] \) denotes the greatest integer \( \le t \). Then \( \displaystyle \int\limits_{0}^{10} f(x)\,dx+\displaystyle \int\limits_{0}^{10} (f(x))^{2}\,dx+\displaystyle \int\limits_{0}^{10} |f(x)|\,dx \) is equal to ________.

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JEE Main 2022 26th June Evening Shift

The integral \( \dfrac{24}{\pi} \displaystyle \int_{0}^{\sqrt{2}} \dfrac{(2 - x^2)\, dx}{(2 + x^2)\sqrt{4 + x^4}} \) is equal to _______.