Monday, May 30, 2016

Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 32

Evaluate t0essin(ts)ds by using Integration by parts.
If we let u=sin(ts) and dv=esds, then
du=cos(ts)dsv=es

So,

t0essin(ts)ds=uvvdu=essin(ts)es(cos(ts))ds=essin(ts)+escos(ts)ds


To evaluate escos(ts)ds, we must use Integration by parts once more...

Hence, if we let u1=cos(ts) and dv1=esds , thendu1=sin(ts)dsv1=es


So, escos(ts)ds=u1v1v1du1=escos(ts)essin(ts)ds

Going back to the first equation,
t0essin(ts)ds=essin(ts)+[escos(ts)essin(ts)ds]

Combining like terms

2t0essin(ts)ds=essin(ts)+escos(ts)t0essin(ts)ds=essin(ts)+escos(ts)2=es2[sin(ts)+cos(ts)]


Evaluating from 0 to t, we have

=et212sint12cost=12(etsintcost)

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