Definite Integration

Let $f^{′}(x)=\frac{192x^{3}}{2+sin^{4}⁡\pi x}$ for all $x\in R$ with $f(\frac{1}{2})=0.$ If $m\leq\int\frac{1}{2}1f(x)dx\leq M,$ then the possible values of m and M are

$$ \text { Match the statements given in Column I with the values given in Column II. } $$

$$ \text { Match the statements/expressions in Column I with the open intervals in Column II. } $$

Match the integrals in Column I with the values in Column II.

Match the following.

The value of the definite integral $\displaystyle\int_{0}^{2} \dfrac{1}{3^{\frac{x+3}{2}}}\, dx$ is

For positive integer $n$, define $f(n)=n+\frac{16+5n-3n^{2}}{4n+3n^{2}}+\frac{32+n-3n^{2}}{8n+3n^{2}}+\frac{48-3n-3n^{2}}{12n+3n^{2}}+\ldots+\frac{25n-7n^{2}}{7n^{2}}$. Then, the value of $limn\to\infty f(n)$ is equal to

Consider the equation$\int_{1}^{e}\frac{(log_{e}x)^{\frac{1}{2}}}{x(a-(log_{e}x)^{\frac{3}{2}})^{2}}dx=1,a\in(-\infty,0)\cup(1,\infty)$.Which of the following statements is/are TRUE?

Let $f:[-\frac{\pi}{2},\frac{\pi}{2}]\toℝ$ be a continuous function such that $f(0)=1$ and $\int_{0}^{\frac{\pi}{3}}f(t)dt=0$Then which of the following statements is (are) TRUE?

Which of the following inequalities is/are TRUE?

Let $f:R\to Randg:R\to R$ be two non-constant differentiable functions. If $f^{′}(x)=(e^{(f(x))-g(x)})g^{′}(x)forallx\in R$, and $f(1)=g(2)=1$ then which of the following statement(s) is (are) TRUE?

If $g(x)=\int_{sinx}^{sin2x}sin^{-1}tdt,$ then