Integral of $Cos(x)dt when $Sin(x)dt is known

[AnjumSKhan]

Lets say I have integral of Cos(x) : $Cos(x) dt = A ^{from 0 to T}, where x is a variable, and A is constant. Now can I find $Sin(x) dt ^{from 0 to T} from this. And vice versa, ie; if Sin and Cos interchange with everything remaining same.

 

[ksdhart]

Okay, so let me see if I understand you correctly. First, it looks like you have two variables t and T. Are those different or are they supposed to be the same? Assuming they're different variables, then the given problem looks something like this:

If \(\displaystyle \int _0^T\cos \left(x\right)dt=A\), find \(\displaystyle \int _0^T\sin \left(x\right)dt\)

Since we're integrating the cosine of x with respect to t, we can treat it as a constant. i.e. \(\displaystyle \int _0^T\cos \left(x\right)dt=\cos \left(x\right)\cdot \left[t\right]^T_0=T\cdot \cos \left(x\right)=A\). Unless, of course, x is meant to be a function of t, in which case I believe we'd need to know how the two variables are related in order to proceed.

 

[AnjumSKhan]

Okay, so let me see if I understand you correctly. First, it looks like you have two variables t and T. Are those different or are they supposed to be the same? Assuming they're different variables, then the given problem looks something like this:

If \(\displaystyle \int _0^T\cos \left(x\right)dt=A\), find \(\displaystyle \int _0^T\sin \left(x\right)dt\)

Since we're integrating the cosine of x with respect to t, we can treat it as a constant. i.e. \(\displaystyle \int _0^T\cos \left(x\right)dt=\cos \left(x\right)\cdot \left[t\right]^T_0=T\cdot \cos \left(x\right)=A\). Unless, of course, x is meant to be a function of t, in which case I believe we'd need to know how the two variables are related in order to proceed.

Hi, thanks for your reply.

At t=0, x=pi/2. At t=T, x=0.