Evaluate the following integral.
$$ \int e^{2x}\sin{5x}\ dx $$
What I have tried :
$ g(x) = \sin5x , f^{'}(x) = e^{2x} , f(x) = e^{2x} $
$$ \int e^{2x}\sin{5x}\ dx = e^{2x}\sin{5x} -\int e^{2x}\ 5\cos{}5x\ dx $$ $$e^{2x}\sin{5x}-5[e^{2x}\cos{5x}- \int e^{2x}-5\ \sin{5x}\ dx]$$ $$e^{2x}\sin{5x}-5[e^{2x}\cos{5x}- 5\int e^{2x}\sin{5x}\ dx]$$
$$\int e^{2x}\sin{5x}\ dx = e^{2x}\sin5x - 5e^{2x}\cos5x + 25\int e^{2x}\sin5x\ dx$$
$$-24\int e^{2x}\sin5x\ dx = e^{2x}\sin5x-5e^{2x}\cos5x$$
$$ \int e^{2x}\sin5x\ dx = \frac{e^{2x}\sin5x-5e^{2x}\cos5x}{-24} $$
The answer in the book is $ \frac{2}{29}e^{2x}\sin5x-\frac{5}{29}e^{2x}\cos5x+C $
Where is my mistake ?
The integral of $e^{2x}$ is, $\int e^{2x}dx = \frac{e^{2x}}{2} + c$. That's your (first) mistake. I didn't look for more.