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

AnjumSKhan

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Feb 11, 2016
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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.
 
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 0Tcos(x)dt=A\displaystyle \int _0^T\cos \left(x\right)dt=A, find 0Tsin(x)dt\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. 0Tcos(x)dt=cos(x)[t]0T=Tcos(x)=A\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.
 
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 0Tcos(x)dt=A\displaystyle \int _0^T\cos \left(x\right)dt=A, find 0Tsin(x)dt\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. 0Tcos(x)dt=cos(x)[t]0T=Tcos(x)=A\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.
 
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