How do integrate #int e^(cos)(t) (sin 2t) dt# between a = 0, #b = pi#?

Answer 1
Start with #int_0^pi e^cos(t)sin(2t) dt = 2int_0^pie^cos(t)sin(t)*cos(t)dt#

Now, by portion

#-2int-sin(t)e^cos(t)*cos(t) dt#
#u'=-sin(t)e^cos(t)# #u=e^cos(t)#
#v = cos(t)# #v' = -sin(t)#
You have : #-2([cos(t)*e^cos(t)]_0^pi - int_0^pi-sin(t)e^cos(t) dt)#
#=-2[cos(t)*e^cos(t)-e^cos(t)]_0^pi#
#=4/e#
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Answer 2

To integrate ( \int_{0}^{\pi} e^{\cos(t)} \sin(2t) , dt ) over the interval ( [0, \pi] ), you can use integration by parts. Let ( u = e^{\cos(t)} ) and ( dv = \sin(2t) , dt ). Then, ( du = -e^{\cos(t)} \sin(t) , dt ) and ( v = -\frac{1}{2}\cos(2t) ).

Applying the integration by parts formula ( \int u , dv = uv - \int v , du ), you'll get:

[ \int_{0}^{\pi} e^{\cos(t)} \sin(2t) , dt = \left[-\frac{1}{2} e^{\cos(t)} \cos(2t)\right]{0}^{\pi} - \int{0}^{\pi} \left(-\frac{1}{2}\cos(2t)\right) (-e^{\cos(t)} \sin(t)) , dt ]

[ = -\frac{1}{2} e^{\cos(\pi)} \cos(2\pi) + \frac{1}{2} e^{\cos(0)} \cos(0) - \frac{1}{2} \int_{0}^{\pi} e^{\cos(t)} \cos(2t) \sin(t) , dt ]

[ = -\frac{1}{2} e^{-1} + \frac{1}{2} - \frac{1}{2} \int_{0}^{\pi} e^{\cos(t)} \cos(2t) \sin(t) , dt ]

The remaining integral might need further manipulation or a different integration technique to solve.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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