I have an integration problem. $$\int \cos(8t) \cdot5e^{5t} dt$$ Integrating such a function by using the integration-by-parts method is complicated and messy. I wanted to use the tabular form, but for some reason, I am missing something, but I don't know what. Could anyone help me to point it out, please?
So far this is what I have:
| Sign | Derivatives | Integration | Product |
|---|---|---|---|
| $+$ | $\cos(8t)$ | $$5e^{5t}$$ | $$\varnothing $$ |
| $-$ | $-8\sin(8t)$ | $$e^{5t}$$ | $$\cos(8t)e^{5t}$$ |
| $+$ | $-64\cos(8t)$ | $$\frac{e^{5t}}{5}$$ | $$\frac{8}{5}\sin(8t)e^{5t}$$ |
| $-$ | $512\sin(8t)$ | $$\frac{e^{5t}}{25}$$ | $$-\frac{64}{5}\int \cos(8t)e^{5t}$$ |
| $+$ | $4096\cos(8t)$ | $$\frac{e^{5t}}{125}$$ | ... |
\begin{align*} 5\int \cos(8t)e^{5t} dt &= \cos(8t)e^{5t}+\frac{8}{5}\sin(8t)e^{5t}-\frac{64}{5}\int \cos(8t)e^{5t}\\ &=\frac{5}{89}\left [ \cos(8t)e^{5t}+\frac{8}{5}\sin(8t)e^{5t} \right]\\ &=\frac{5\cos(8t)e^{5t}+8\sin(8t)e^{5t}}{89} \end{align*}
But this is not the correct answer. The correct answer is: $$\frac{25\cos(8t)e^{5t}+40\sin(8t)e^{5t}}{89}$$ Please help me out here.
With tabular integration, products of diagonal entries lie outside of the integral, and the product of the entries in the final row of the completed table lies inside our integral. (In your calculation, it appears you have no such "horizontal product" and instead the integral at the end is simply the final diagonal product.)
The DI table I would get is
which gives the solution as follows (if we're careful to multiply the inside of the final integral by $5$ and divide on the outside by $5$ so we have the same integral as we start with):
$$\int 5e^{5t} \cos(8t) \, \mathrm{d} t = \color{blue}{e^{5t} \cos(8t) + \frac 8 5 e^{5t} \sin(8t)} \color{green}{ -\frac{64}{25} \int 5 e^{5t} \cos(8t) \, \mathrm{d}t}$$
Of course, if our original integral is $\mathcal{I}$, this amounts to the equation
$$5\mathcal{I} = e^{5t} \cos(8t) + \frac 8 5 e^{5t} \sin(8t) - \frac{64}{25} \mathcal{I}$$
and so, solving for $\mathcal{I}$
$$\mathcal{I} = \frac{25e^{5t}}{89} \left( \cos(8t) + \frac 8 5 \sin(8t) \right) + C$$
As a general look, the DI method would imply, for the below table,
that
$$\int a(x)f(x) \, \mathrm{d} x = \color{blue}{ a(x) \cdot g(x) - b(x) \cdot h(x)} \color{green}{ + \int c(x) \cdot h(x) \, \mathrm{d}x}$$
Notice that the items on the diagonals (marked in blue) are multiplied together and not integrated, but we have to tack on an integral: the integral of the product of the entries in the final row (not diagonally opposed, and marked in green).