List of top Mathematics Questions on Integral Calculus asked in JEE Main

If $\int \frac{dx}{(x^2 + x + 1)^2} = a \tan^{-1} \left( \frac{2x + 1}{\sqrt{3}} \right) + b \left( \frac{2x + 1}{x^2 + x + 1} \right) + C, x>0$ where C is the constant of integration, then the value of $9(\sqrt{3}a + b)$ is equal to _________.
If \( \int \frac{2e^x + 3e^{-x}}{4e^x + 7e^{-x}} dx = \frac{1}{14}(ux + v \log_e(4e^x + 7e^{-x})) + C \), where C is a constant of integration, then u + v is equal to _________.
\( \displaystyle \lim_{x \to 0} \frac{\int_{0}^{x^2} \sin(\sqrt{t}) \, dt}{x^3} \) is equal to :
If f(x) = ∫$_{x}^{1}$ (5x⁸ + 7x⁶)/(x² + 1 + 2x⁷)² dx, (x ≥ 0), f(0) = 0 and f(1) = 1/K, then the value of K is __________.