[Security to the Party 27: Conspiracy Edition]

Question

You're browsing Puzzling.SE when suddenly you uncover a massive conspiracy. The posters of the "Security to the Party" questions are just posting 7 or 8 random numbers and then accepting the second or third answer which explains the data.

Infuriated, you attempt to break into the Dark Puzzlers Chat. However, there is a guard, et cetera.

A puzzler arrives. The guard says 1 5 2 1 4 3 8, and she replies 10 2 6.

A second puzzler appears. The guard says CODREEF, and he replies PSF.

A third puzzler arrives. The guard says 1.3 2.0 1.6 1.8 2.2 2.3 2.4, and the puzzler replies 2.0 2.6 2.6.

For the fourth puzzler, the guard says 6, 3x^2+1, 6x+8, 6(x+6)^2, (x+1)(3x+5), sin(x)/x, (x+1)^2(x+2). The puzzler complains that the last one is impossible and leaves in a huff.

Obviously, you've heard enough, because anyone can work out any pattern after four examples. You approach the guard.

The guard says Bleach as needed, Doctor Whooves's cutie mark, Play video, Infinity, Give way to traffic, Cancer horoscope, Normal subgroup. What should you reply?

Answer 1

Bowtie, 8, 69.
The 1st, 3rd, 5th, and 7th things that the guard says are derived from each other by the same operation. The answer is to apply that operation on the 2nd, 4th, and 6th things.
"Bleach as Needed" is a triangle pointing up, "Play video" is a triangle pointing to the right, "Give way to traffic" is a triangle pointing down, and "Normal subgroup" is a triangle pointing to the left. Therefore the solution is to turn the other three objects 90 degrees clockwise.
First puzzler: double the numbers.
Second puzzler: take the next letter of the alphabet.
Third puzzler: add the decimal part of the number to the number.
Fourth puzzler: provide an antiderivative.

Answer 2

Also, the fourth puzzler, who left in a huff, should have answered (according to user5352):

$x^3+x+c,\ 2x^3+36x^2+216x+c,\ Si(x),$ where $Si(x)$ is the sine integral

To solve the antiderivative of $x^2+1$, we can have

\begin{align} & \quad \int x^2+1\ dx\\ &=\frac{x^{2+1}}{2+1}+\frac{x^{0+1}}{0+1}+c\\ &=\frac{x^3}{3}+x+c \end{align}

To solve the antiderivative of $6(x+6)^2$, we can have

\begin{align} & \quad \int 6(x+6)^2\ dx\\ &=\int6(x^2+12x+36)\ dx\\ &=\int6x^2+72x+216\ dx\\ &=\frac{6x^{2+1}}{2+1}+\frac{72x^{1+1}}{1+1}+\frac{216x^{0+1}}{0+1}+c\\ &=\frac{6x^3}{3}+\frac{72x^2}{2}+216x+c\\ &=2x^3+36x^2+216x+c \end{align}

Now, as the fourth puzzler said, the last one is impossible - but there is a name for the antiderivative of $\dfrac{\sin x}{x}$:

$Si(x)$, or the sine integral

We have now finished the process of finding out what the fourth puzzler should have said.