Question

What is sinx integral
You will be marked as brainliest answer by Ujjwalchauhan07

Answer 1

Answer:

Step-by-step explanation:

Well, to determine the antiderivative of sinx, you have to know your limits, and, in fact, your derivatives.

Let me explain:

d(sinx)dx=limϵ→0sin(x+ϵ)−sinxϵ

But we have

sin(x+ϵ)−sinx=2cos(x+ϵ+x2)sin(x+ϵ−x2)=2cos(x+ϵ2)sin(ϵ2)

So

d(sinx)dx=limϵ→02cos(x+ϵ2)sin(ϵ2ϵ=(limϵ→0cos(x+ϵ2))(limϵ→0sinϵ2ϵ2)

As

limx→0sinxx=1

d(sinx)dx=limϵ→0cos(x+ϵ2)=cosx

Which might give us a clue. Let’s look at

d(cosx)dx=limϵ→0cos(x+ϵ)−cosxϵ

We have

cos(x+ϵ)−cosx=2sin(x+ϵ+x2)sinx−(x+ϵ)2=2sin(x+ϵ2)sin(−ϵ2)=−2sin(x+ϵ2)sinϵ2

So

d(cosx)dx=limϵ→0−2sin(x+ϵ2)sinϵ2ϵ=−(limϵ→0sin(x+ϵ2))(limϵ→0sinϵ2ϵ2)

We still have

limx→0sinxx=1

So

d(cosx)dx=−limϵ→0sin(x+ϵ2)=−sinx

So

∫sinxdx=−cosx