Definite Integrals

Let the domain of the function f(x) = log₂(log₄(log₆(3 + 4x - x²))) be (a, b). If ∫ₐᵇ [x²]dx = p - √q - √r, p, q, r ∈ N, gcd(p, q, r) = 1, where [·] is the greatest integer function, then p + q + r is equal to

Let [t] denote the greatest integer less than or equal to t. Then the value of ∫?²|2x-[3x]|dx is

The region represented by {z = x+iy∈C: |z|-Re(z)≤1} is also given by the inequality:

The value of the definite integral ∫π/?π/? dx / (1 + ³√tan2x) is:

The value of the definite integral ∫π/?π/? dx / (1 + ³√tan2x) is

If ${\theta }_1$ and ${\theta }_2$ be respectively the smallest and the largest values of $\theta$ in $\left(\mathrm{0,2}\pi \right)-\left\{\pi \right\}$ which satisfy the equation, $2{\mathrm{c}\mathrm{o}\mathrm{t}}^2\theta -\frac{5}{\mathrm{s}\mathrm{i}\mathrm{n}\theta }+4=0$ , then ${\int }_{{\theta }_1}^{{\theta }_2}?{\mathrm{c}\mathrm{o}\mathrm{s}}^2?3\theta d\theta$ is equal to:

The value of the integral ∫₋₁¹ log(x + √(x² + 1))dx is: