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Let \( f(x)=\int \dfrac{2x}{(x^{2}+1)(x^{2}+3)}\,dx \). If \( f(3)=\dfrac{1}{2}\big(\log_{e}5-\log_{e}6\big) \), then \( f(4) \) is equal to:

Let \( I(x)=\displaystyle\int \sqrt{\frac{x+7}{x}}\,dx \) 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 ____.

If \( I(x) =\displaystyle \int e^{\sin^2 x}\,(\cos x \sin 2x - \sin x)\,dx \) and \( I(0) = 1 \), then \( I\!\left(\dfrac{\pi}{3}\right) \) is equal to:

For \( \alpha, \beta, \gamma, \delta \in \mathbb{N} \), if \( \displaystyle \int \left( \left( \dfrac{x}{e} \right)^{2x} + \left( \dfrac{e}{x} \right)^{2x} \right) \log_e x \, dx = \dfrac{1}{\alpha} \left( \dfrac{e}{x} \right)^{\beta x} - \dfrac{1}{\gamma} \left( \dfrac{e}{x} \right)^{\delta x} + C \), where \( e=\displaystyle \sum_{n=0}^{\infty}\dfrac{1}{n!} \) and \( C \) is constant of integration, then \( \alpha+2\beta+3\gamma-4\delta \) is equal to:

Let \( I(x) = \displaystyle \int \frac{(x+1)}{x(1 + x e^{x})^{2}}\,dx,\; x > 0. \) If \( \displaystyle \lim_{x \to \infty} I(x) = 0 \), then \( I(1) \) is equal to:

If \(\displaystyle \int_{-0.15}^{0.15} \left|100x^2-1\right| dx=\frac{k}{3000} \), then \( k \) is equal to ____.

Let \( I(x) = \int \dfrac{x^{2}\left(x\sec^{2}x + \tan x\right)}{(x \tan x + 1)^{2}} \, dx \). If \( I(0) = 0 \), then \( I\!\left(\dfrac{\pi}{4}\right) \) is equal to:

If $\int \sqrt{\sec 2x - 1} \, dx = \alpha \log_e \left| \cos 2x + \beta + \sqrt{\cos 2x \left( 1 + \cos \frac{1}{\beta}x \right)} \right| + \text{constant}$, then $\beta - \alpha$ is equal to ________.

Let \( f(x)=\displaystyle\int\limits \dfrac{dx}{(3+4x^2)\sqrt{4-3x^2}},\ |x|<\frac{2}{\sqrt{3}} \). If \( f(0)=0 \) and \( f(1)=\dfrac{1}{\alpha\beta}\tan^{-1}\!\left(\dfrac{\alpha}{\beta}\right),\ \alpha,\beta>0 \), then \( \alpha^2+\beta^2 \) is equal to __________.

Let \( \alpha \) be the area of the larger region bounded by the curve \( y^{2} = 8x \) and the \( y = x \) and \( x = 2 \), which lies in the first quadrant. Then the value of \( 3\alpha \) is equal to ________.

The integral \( \int\!\left(\left(\dfrac{x}{2}\right)^{x}+\left(\dfrac{2}{x}\right)^{x}\right)\log_{2}x\,dx \) is equal to:

Let \( f(x) \) be a function satisfying \( f(x) + f(\pi - x) = \pi^2, \forall x \in \mathbb{R} \). Then \(\displaystyle \int_{0}^{\pi} f(x)\sin x \, dx \) is equal to:

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JEE Main 2023 12th April Morning Shift

Let \( I(x)=\displaystyle\int \sqrt{\frac{x+7}{x}}\,dx \) 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 ____.