Maths Integrals

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

If S and S' are the foci of the ellipse x²/18 + y²/9 = 1 and P be a point on the ellipse, then min(SP⋅S'P) + max(SP⋅S'P) is equal to :

The integral ∫₀² x-1|-x|dx is equal to [numerical value].1|-x|dx is equal to.

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

Let f: [0,∞) → [0,∞) be defined as f(x) = ∫? [y]dy where [x] is the greatest integer less than or equal to x. Which of the following is true?

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:

If $\frac{}{{}^{}{\left({}^{}\right)}^{}}{\left({}^{}\right)}^{}$ $\int \frac{\mathrm{c}\mathrm{o}\mathrm{s}?xdx}{{\mathrm{s}\mathrm{i}\mathrm{n}}^3?x{\left(1+{\mathrm{s}\mathrm{i}\mathrm{n}}^6?x\right)}^{2}3}=f\left(x\right){\left(1+{\mathrm{s}\mathrm{i}\mathrm{n}}^6?x\right)}^{1/\lambda }+c$ where $c$ $$ is a constant of integration, then $\lambda f\left(\frac{\pi }{3}\right)$ $\left(\frac{}{}\right)$ is equal to: