Grade 12Integral Calculus

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

Question image for what will be the integration of e to the power x

Sumit Jha

7 Years agoGrade 12

1 Answer

Arun

7 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 +