List of top Mathematics Questions on Some Properties of Definite Integrals asked in WBJEE

The value of \( \displaystyle \int_0^{1.5} \left\lfloor x^2 \right\rfloor \, dx \) is:
The value of \( \int_{-100}^{100} \frac{x + x^3 + x^5}{1 + x^2 + x^4 + x^6} dx \) is:
The value of \(\int_{0}^{\frac{1}{2}}\frac{dx}{\sqrt{1-x^{2n}}}\) is (n∈N)
The expression \(\frac{\int_{0}^{n}[x]dx}{\int_{0}^{n}\left \{ x\right \}dx}\), where [x] and {x} are respectively integral and fractional part of x and n∈N, is equal to
The value of $\int_0^{\pi/2} \frac{(\cos x)\sin x}{(\cos x)\sin x + (\sin x)\cos x} \, dx$ is
Let $f$ be a non-negative function defined in $[0, \pi/2]$, $f'$ exists and is continuous for all $x$, and $\int_0^x \sqrt{1 - (f'(t))^2} dt = \int_0^x f(t) dt$ and $f(0) = 0$. Then
If $I$ is the greatest of $I_1 = \int_0^1 e^{-x} \cos^2 x \, dx$, $I_2 = \int_0^1 e^{-x^2} \cos^2 x \, dx$, $I_3 = \int_0^1 e^{-x^2} \, dx$, $I_4 = \int_0^1 e^{-\frac{x^2}{2}} \, dx$, then
The value of the integral $I = \int\limits^{2014}_{1/2014} \frac{\tan^{-1} x}{x} dx $ is