# Integral of e^(2x)sin[3x]?

1. Apr 20, 2005

### kennis2

Integral of e^(2x)sin[3x]??

Integration by parts of: e^(2x)sin[3x]
result:e^2x(2sin3x-3cos3x)/13+C
can't get that result =(

2. Apr 20, 2005

### HallsofIvy

Staff Emeritus
What have you DONE? Show us what you have done and we may be able to point out your mistakes.

3. Apr 20, 2005

### Jameson

Right, you do need to show your work. But I'll give you a hint. You're going to have to do integration by parts twice. Post what happens when you do this and you'll see how this problem can be manipulated to get the correct answer.

4. Apr 20, 2005

### Theelectricchild

Bah integration by parts is a long and tedious process... id express the sin 3x term as complex exponentials and work from there...

you would do well the recall that

$$sin({\omega}t)=\frac{e^{j{\omega}t}-e^{-j{\omega}t}}{2j}$$

5. Apr 20, 2005

### Jameson

I would consider that more difficult, but to each his own.

6. Apr 20, 2005

### dextercioby

Here's a very elegant way

$$\int e^{2x}\sin 3x \ dx=\mbox{Im}\left(\int e^{(2+3i)x} \ dx\right)=\mbox{Im}\left(\frac{e^{2x+3ix}}{2+3i}\right)+C=...=\frac{2\sin 3x-3\cos 3x}{13}e^{2x}+C$$

1.I'll let u do the intermediate calculations...
2.Excercise:compute

$$\int e^{-x}\cos 4x \ dx$$

Daniel.

7. Apr 26, 2005

### Hippo

Daniel, I can compute that in a really awkward way, expressing cos(4x) with complex exponentials.

$$\int e^{-x}\cos 4x \ dx = (\frac{1}{2}) \int e^{x(4j-1)} + e^{x(-4j-1)} \ dx$$

$$= \frac{(-4j-1)e^{x(4j-1)} + (4j-1)e^{x(-4j-1)}}{34} + K$$

How did you do the original integral, though?

[Edit: Yeah, I accidentally edited this instead of quoting it. I had to rewrite it.]

Last edited by a moderator: Apr 27, 2005
8. Apr 27, 2005

### whozum

Its probably an example of integration by parts in his book, Jameson had the best hint, IMO as it was the one provided in my textbook.

9. Apr 27, 2005

### dextercioby

Daniel.

10. Apr 27, 2005

### Hippo

Look again.

$$\frac {e^{x(4j-1)}} {2(4j-1)} + \frac{e^{x(-4j-1)}}{2(-4j-1)} + K = \frac { (-4j-1) e^{x(4j-1)} + (4j-1) e^{x(-4j-1)} }{2*17} + K$$

Now, will you please tell me how you did the original integral? I don't understand your method or notation.

Last edited by a moderator: Apr 27, 2005
11. Apr 28, 2005

### dextercioby

$$\cos 4x=\frac{1}{2}\left(e^{4ix}+e^{-4ix}\right)$$

$$e^{-x}\cos 4x=\frac{1}{2}\left[e^{x(4i-1)}+e^{x(-4i-1)}\right]$$

$$\int e^{-x}\cos 4x \ dx=\frac{1}{2}\left[\int e^{x(4i-1)} \ dx+\int e^{x(-4i-1)} \ dx\right]=\frac{1}{2}\left[\frac{e^{x(4i-1)}}{4i-1}+\frac{e^{x(-4i-1)}}{-4i-1}\right]+C$$

$$=\frac{1}{2}\left\{\frac{1}{17}\left[(-4i-1)e^{x(4i-1)}+(4i-1)e^{x(-4i-1)}\right]\right\}+C=\frac{1}{17}\left(4\sin 4x-\cos 4x\right)e^{-x} +C$$

Daniel.

12. Apr 28, 2005

### Hippo

Yeah, I guess I could've expressed my answer in a better form. :grumpy:

13. Apr 28, 2005

### dextercioby

The algebra was pretty simple,just be careful with prducts of "i"-s and "-1"-s...

Daniel.

14. Apr 30, 2005

### SplinterIon

Excuse my ignorance (I'm only a highschool student), but how did you get from

$$\int e^{2x}\sin 3x \ dx=\mbox{Im}\left(\int e^{(2+3i)x} \ dx\right)$$

My knowledge of complex numbers is limited and I tried applying euler's formula/de moivre's theorem and got nowhere. Even working backwards doesn't seem to help. I was however able to do integration by parts without sweat.

15. Apr 30, 2005

### p53ud0 dr34m5

euler's formula
$$e^{ix}=cos(x)+isin(x)$$

or some variation of that.

16. May 1, 2005

### µ³

$$\int e^{2x}sin(3x) dx = -\frac{1}{3}e^{2x}cos(3x) - \frac{2}{3}\int cos(3x) e^{2x} dx$$
$$\int e^{2x}sin(3x) dx = -\frac{1}{3}e^{2x}cos(3x) - \frac{2}{3}(\frac{1}{3}e^{2x}sin(3x) - \frac{2}{3}\int e^{2x} sin(3x) dx)$$
Solve for
$$\int e^{2x}sin(3x) dx$$

Disclaimer: It's 12:46am here and I didn't bother to do this on paper, some negatives or fractions maybe wrong but the approach is correct: seperate by parts until you get an equation that you can solve for the original integral.

17. May 2, 2005

### Hessam

why not just simply to tabular integration twice?

integral of e^2x * sin3x

original terms followed by deriv's/anti-deriv's

e^2x ------- sin3x
2e^2x ------ (-cos3x)/3
4e^2x ------ (-sin3x)/9

which then yields

=-(cos3x * e^2x)/3 + (2/9)(e^2x * sin3x) - (4/9) ( integral of e^2x * sin3x)

now take that last integral, add it to both sides , thus leaving

(9/13)[-(cos3x * e^2x)/3 + (2/9)(e^2x * sin3x)] + C