Verified answer

let u=e^(2x)

du=2e^(2x)

let dv=sin(5x) dx

v=-1/5cos(5x)

∫(e^(2x))(sin(5x))dx=-1/5cos(5x)e^(2x)-∫-2/5cos(5x)e^(2x) dx

∫-2/5cos(5x)e^(2x) dx=-2/5∫cos(5x)e^(2x) dx

u=e^(2x)

du=2e^(2x)

let dw=cos(5x) dx

w=1/5sin(5x)

∫cos(5x)e^(2x)=1/5e^(2x)sin(5x)-∫2/5sin(5x)e^(2x)

∫(e^(2x))(sin(5x))dx=-1/5cos(5x)e^(2x)+2/5∫cos(5x)e^(2x) dx

∫(e^(2x))(sin(5x))dx=-1/5cos(5x)e^(2x)+(2/5)(1/5e^(2x)sin(5x)-2/5∫sin(5x)e^(2x) dx)

let I=∫e^(2x)sin(5x) dx

I=-1/5cos(5x)e^(2x) + (2/5)(1/5e^(2x)sin(5x)-2/5I

7/5I=-1/5cos(5x)e^(2x) + (2/5)(1/5e^(2x)sin(5x)

I=-1/7cos(5x)e^(2x) + (35/2)e^(2x)sin(5x)+C

Integration by parts is explained with a link in the source. It is much easier to do in terms of functions and derivatives rather than differencials.

Integrate the original integrand by parts:

∫ e²˟sin(5x) dx

Let f'(x) = e²˟

f(x) = e²˟ / 2

Let g(x) = sin(5x)

g'(x) = 5cos(5x)

∫ f'(x)g(x) dx = f(x)g(x) - ∫ f(x)g'(x) dx

∫ e²˟sin(5x) dx = e²˟sin(5x) / 2 - ∫ 5e²˟cos(5x) / 2 dx

Integrate the new integrand by parts:

∫ 5e²˟cos(5x) / 2 dx

Let f'(x) = e²˟ / 2

f(x) = e²˟ / 4

Let g(x) = 5cos(5x)

g'(x) = -25sin(5x)

∫ f'(x)g(x) dx = f(x)g(x) - ∫ f(x)g'(x) dx

∫ 5e²˟cos(5x) / 2 dx = 5e²˟cos(5x) / 4 + ∫ 25e²˟sin(5x) / 4 dx

Put it all together and collect like terms to solve for the integral:

∫ e²˟sin(5x) dx = e²˟sin(5x) / 2 - ∫ 5e²˟cos(5x) / 2 dx

∫ e²˟sin(5x) dx = e²˟sin(5x) / 2 - [5e²˟cos(5x) / 4 + ∫ 25e²˟sin(5x) / 4 dx]

∫ e²˟sin(5x) dx = e²˟sin(5x) / 2 - 5e²˟cos(5x) / 4 - ∫ 25e²˟sin(5x) / 4 dx

∫ e²˟sin(5x) dx = e²˟sin(5x) / 2 - 5e²˟cos(5x) / 4 - ∫ 25e²˟sin(5x) / 4 dx

29 ∫ e²˟sin(5x) dx / 4 = e²˟sin(5x) / 2 - 5e²˟cos(5x) / 4

∫ e²˟sin(5x) dx = [2e²˟sin(5x) - 5e²˟cos(5x)] / 29 + C

integrate by parts

i.e. =integral(e^(2x) d/(dx) (-1/5 cos(5x)) dx=-1/5cos(5x) e^(2x) + 2/5 integral(e^(2x)cos(5x))

integral(e^(2x)cos(5x))=integral(e^(2x) d/(dx) (1/5)sin(5x))=1/5sin(5x) e^(2x) - 2/5 integral(e^(2x)sin(5x))

therefore integral (e^(2x))(sin(5x))dx=-1/5cos(5x) e^(2x) +2/5 (1/5sin(5x) e^(2x) - 2/5 integral(e^(2x)sin(5x)) )= 2/25 sin(5x)e^(2x) -1/5cos(5x) e^(2x) - 4/25 integral(e^(2x)sin(5x))

therefore 29/25 integral(e^(2x)sin(5x))=2/25 sin(5x)e^(2x) -1/5cos(5x) e^(2x)

therefore integral(e^(2x)sin(5x))=25/29 (2/25 sin(5x)e^(2x) -1/5cos(5x) e^(2x)) = 2/29 sin(5x)e^(2x) - 5/29 cos(5x) e^(2x)