Integrating e^(2 theta)sin(9 theta) d theta

[tangibleLime]

Homework Statement

Integrate.

$$\int e^{2\theta}sin(9\theta) d\theta$$​

Homework Equations

Integration by Parts

$$\int udv = uv - \int vdu$$​

The Attempt at a Solution

I started out with integration by parts (IPB).

$$u = e^{2\theta}$$

$$du = 2e^{2\theta} d\theta$$

$$v = \frac{-1}{9}cos(9\theta)$$

$$dv = sin(9\theta) d\theta$$

$$uv - \int vdu = -\frac{1}{9}cos(9\theta)*e^{2\theta}+\frac{2}{9}\int cos(9\theta) d\theta$$​

Noticing that the new integral on the right is just about the same as how I started (sin just turned into cos), I did IPB once again.

$$u = e^{2\theta}$$

$$du = 2e^{2\theta} d\theta$$

$$v = \frac{1}{9}sin(9\theta}$$

$$dv = cos(9\theta) d\theta$$

$$uv - \int vdu = \frac{1}{9}sin(9\theta)*e^{2\theta}-\frac{2}{9}\int sin(9\theta)*e^{2\theta} d\theta$$​

Combining all of this into the equation so far...

$$-\frac{1}{9}cos(9\theta)*e^{2\theta}+\frac{2}{81}sin(9\theta)*e^{2\theta}-\frac{2}{9}\int sin(9\theta)*e^{2\theta} d\theta$$​

Noticing that this new integral on the right is the exactly equal to what I started with, I added it to both sides in order to remove it from the right side. I mentally noted that the left version could have a coefficient of $$\frac{9}{9}$$, and added the right side's coefficient of $$\frac{2}{9}$$ to give me the final coefficient for the left version of $$\frac{11}{9}$$.

$$\frac{11}{9}\int e^{2\theta}sin(9\theta) d\theta = -\frac{1}{9}cos(9\theta)e^{2\theta}+\frac{2}{81}sin(9\theta)e^{2\theta}$$​

To get that $$\frac{11}{9}\$$ out of the left side of the equation to get my original integral back, I multiplied both sides by it's reciprocal, $$\frac{9}{11}$$. Then I tacked on the constant of integration to arrive at my final answer.

$$\int e^{2\theta}sin(9\theta) d\theta = \frac{9}{11}(-\frac{1}{9}cos(9\theta)e^{2\theta}+\frac{2}{81}sin(9\theta)e^{2\theta}) + C$$
​

This was marked wrong, even after I simplified the answer (which I did not do here, but I checked the simplification with WolframAlpha). Any suggestions?​

 

[vela]

If you differentiate your result, you get 85/99 e^2tsin 9t, which is almost what you started with, so it looks like you just made an arithmetic or algebra error somewhere. Your method is correct.

 

[tangibleLime]

Thanks, I figured as much, that's usually my downfall :D As long as I know my method is correct, I'm okay with it.