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what will be the integration of e to the power x sin square x dx?

what will be the  integration of e to the power x sin square x dx?

Question Image
Grade:12

1 Answers

Arun
25763 Points
2 years ago
Let L = ∫ e^x*sin(2x) dx and use integration by parts. 

Let: 

u = sin(2x) 
du = 2cos(2x) dx 
dv = e^x dx 
v = e^x 

Then: 

L = uv - ∫ v du 
==> L = e^x*sin(2x) - 2 ∫ e^x*cos(2x) dx 

Let: 

u = cos(2x) 
du = -2sin(2x) 
dv = e^x dx 
v = e^x 

By another round of integration by parts: 

L = e^x*sin(2x) - 2(uv - ∫ v du) 
==> L = e^x*sin(2x) - 2[e^x*cos(2x) + 2 ∫ e^x*sin(2x) dx] 
==> L = e^x*sin(2x) - 2e^x*cos(2x) - 4 ∫ e^x*sin(2x) dx 
==> L = e^x*sin(2x) - 2e^x*cos(2x) - 4L (since L = ∫ e^x*sin(2x) dx) 
==> 5L = e^x*sin(2x) - 2e^x*cos(2x) 
==> 5L = e^x * [sin(2x) - 2cos(2x)] 
==> L = e^x * [sin(2x) - 2cos(2x)]/5 
==> ∫ e^x*sin(2x) dx = e^x * [sin(2x) - 2cos(2x)]/5 + C 

Therefore, ∫ e^x*sin(2x) dx = e^x * [sin(2x) - 2cos(2x)]/5 +

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