0

I was wondering if the integral of $\sin\left(\int(\sin(x)dx\right)$ is equivalent to 0, if it had an answer, or if there was simply no answer?

amWhy
  • 209,954
Kempton Osborn
  • 1
  • Do you know the relation between derivatives, antiderivatives and integrals ? –  May 21 '17 at 16:59
  • What does integral mean? The integral is defined as the antiderivative, therefore how can $\int\sin xdx=0$ – Teh Rod May 21 '17 at 16:59
  • I think the OP is asking what $\sin ( \int \sin x dx)$ is, and is somehow wondering if $f(\int f(x) dx)$ will "cancel each other out". That answer is, no, an integral is not an "opposite". Since $\int \sin x dx = -\cos x + C$ then$\sin (\int \sin x dx) = \sin (-\cos x + C) = -\sin (\cos x)\cos C + \sin C\cos(\cos x)$ (if we assume $C = 0$ this is simply $-\sin (\cos x)$. – fleablood May 21 '17 at 17:19

1 Answers1

0

Why would it be zero or unknowable?

$\int \sin x dx = -\cos x + c$ so $\sin (\int \sin x dx) = \sin (-\cos x + c)$

$= \sin(-\cos x)\cos c + \sin c \cos(-\cos x) $

$= - \sin(\cos x)\cos c + \sin c \cos(\cos x)$.

fleablood
  • 124,253