$1+3+5^2+7+9^2+\ldots$ upto 40 terms is equal to

If $10 \sin ^4 \theta+15 \cos ^4 \theta=6$, then the value of $\frac{27 \operatorname{cosec}^6 \theta+8 \sec ^6 \theta}{16 \sec ^8 \theta}$ is

The value of $\int_\limits{-1}^1 \frac{(1+\sqrt{|x|-x}) e^x+(\sqrt{|x|-x}) e^{-x}}{e^x+e^{-x}} d x$ is equal to

Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that

$f(x)=1-2 x+\int_0^x e^{x-t} f(t) d t$ for all $x \in[0, \infty)$.

Then the area of the region bounded by $y=f(x)$ and the coordinate axes is