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:
\( \log_{e}17-\log_{e}18 \)
\( \dfrac{1}{2}\big(\log_{e}19-\log_{e}17\big) \)
\( \dfrac{1}{2}\big(\log_{e}17-\log_{e}19\big) \)
\( \log_{e}19-\log_{e}20 \)
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:
