Find the value of integral in the branch b?

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1 Answer
May 24, 2018

= 1/2( e sin 1 - e cos 1 + 1)

Explanation:

I = int_1^e sin ( ln y ) dy

Let:

  • ln y = x, qquad y = e^x, qquad 1/y dy = dx

I = int_0^1 sin ( x ) e^x dx

Take the first result:

  • int e^x sin x \ dx = 1/2 e^x ( sin x - cos x) + C

I = ( 1/2 e^x ( sin x - cos x) )_0^1

= ( 1/2 e ( sin 1 - cos 1) ) - ( 1/2 e^0 ( sin 0 - cos 0) )

= 1/2( e sin 1 - e cos 1 + 1)