PYQs Based Test - 04 : Definite and Indefinite Integrals

Let x be a continuous variable defined over the interval (-∞, ∞) and $f\left(x\right)=e^{-x-e^{-x}}$. The integral $g\left(x\right)=\int f\left(x\right)dx$ is equal to

Consider a spatial curve in three-dimensional space given in parametric form by $X\left(t\right)=\cos t,y\left(t\right)=\sin t,z\left(t\right)=\frac{2}{\pi }t,0\le t\le \frac{\pi }{2}$

The length of the curve is ______

If for non-zero x, if $af\left(x\right)+bf\left(\frac{1}{x}\right)=\frac{1}{x}-25$ where a ≠ b then $\underset{1}{\overset{2}{\int }}f\left(x\right)dx$ is

Which of the following is NOT true for all possible non-zero choices of integers m, n; m ≠ n, or all possible non-zero choices of real numbers p, q; p ≠ q, as applicable?

The value of the definite integral $\underset{1}{\overset{e}{\int }}\sqrt{x}\ln \left(x\right)dx$ is