How to integrate $\int e^{-x}\sin(3x)\;dx$?

I want to integrate following by the method of integration by parts

$$\frac{\cos(3x)}{e^{x}}$$

when I try to solve it by integration by parts it always leads to something like as mentioned below and it still have integration sign around it.

$$\frac{\sin(3x)}{e^{x}}$$

• Any reason you are dividing by $e^{-x}$ instead of multiplying by $e^x$? Also: You can find some good starting points on how to format mathematics on the site here. This AMS reference is very useful. – Zev Chonoles Jun 8 '13 at 10:36
• just bring $e^{-x}$ to numerator. You will be able to do it with integration by parts. You have to do it twice. – Maesumi Jun 8 '13 at 10:36
• math.stackexchange.com/questions/408515/liate-ilate-rule/… try this(check my answer) – iostream007 Jun 8 '13 at 10:38
• Is the present form perhaps the question that you intended to ask? – Lord_Farin Jun 8 '13 at 10:45
• proofwiki.org/wiki/… – lab bhattacharjee Jun 8 '13 at 10:47

we have $\int \sin(3x)e^{-x}dx$ so we take $f=\sin(3x),g'=e^{-x}$ getting $$=-\sin(3x)e^{-x}+3\int \cos(3x)e^{-x}dx$$

and integrating by parts again ($f=\cos(3x),g'=e^{-x}$) gives $$\int \cos(3x)e^{-x}dx=-\cos(3x)e^{-x}-3\int \sin(3x)e^{-x}dx$$ or generally we can conclude $$\int \sin(3x)e^{-x}=-\sin(3x)e^{-x}-3\cos(3x)e^{-x}-9\int \sin(3x)e^{-x} dx$$ and then that's clear that (if I didn't have any calculation mistake):$$\int \sin(3x)e^{-x}dx=\frac {-3\cos(3x)e^{-x}-\sin(3x)e^{-x}}{10}+C$$

• There were some presentational issues; the integrand is actually $\sin(3x)e^x$. Also, \sin (resp. \cos) in math mode gives $\sin$ ($\cos$), which is better than $sin$ ($cos$). – Lord_Farin Jun 8 '13 at 10:49
• @DanielMargolis: Your edit was correct. Unfortunately, it was mistakenly rejected. I have made your changes, but there is no way to get you your points for the suggested edit. – robjohn Jun 8 '13 at 12:19

$$\int e^{3ix}e^{-x}dx=\int e^{(3i-1)x}dx=\frac{1}{3i-1}e^{(3i-1)x}+C=-\frac{1+3i}{10}e^{-x}(\cos(3x)+i\sin(3x))+C$$ Now expand and take the imaginary part.

• Now... why do we take the imaginary part? – apnorton Jun 8 '13 at 11:42
• @anorton Because you are integrating $\sin$ and $\sin$ is the imaginary part of this when you expand it. – Kaish Jun 8 '13 at 11:59
• @anorton If it helps looking at it this way, $\int \Im(f(x))dx=\Im(\int f(x)dx)$. – Meow Jun 8 '13 at 12:21

The way I write integration by parts is by first choosing one of the functions to be $d$, which is the one we differentiate, and the other function to be $i$, which is the one we integrate. Then, we integrate by parts by doing the following: $d \times \int i - \int \{\int i \cdot$ differential of $d \}$. Which, I think is the same as the formula, but I don't really know how to use the formula properly because I was taught to think of it in this way. Now for your integral, we start of by saying let

$$I =\int e^{-x} \sin(3x)dx$$

If we select $e^{-x}$ to be the one we integrate and so we will differentiate $\sin(3x)$, as that is easier. So now, if we apply my by parts algorithm to calculate $I$, we start off by holding $\sin(3x)$, multiplying it by $\int e^{-x}dx = -e^{-x}$, and then subtracting the integral of $-\int e^{-x} \cdot 3 \cos(3x)$, so we get

$$I = \sin(3x) \cdot - e^{-x} - \int \left( - \int e^{-x} \cdot 3 \cdot \cos(3x) \right)$$ $$= - \sin(3x) e^{-x} - 3 \int \left(e^{-x} \cos(3x)\right)dx.$$

Now, once again, we apply the by parts algorithm to the new bit involving $\cos$. Again, we can choose to integrate $e^{-x}$ and differentiate $\cos(3x)$ and so we get

$$I = - \sin(3x)e^{-x} - 3 \left[ \cos(3x) \cdot -e^{-x} - \int \left( - \int e^{-x} \cdot - 3\sin(3x) \right) \right]$$

tidying this up gives us

$$I = - \sin(3x)e^{-x} - 3 \left[-\cos(3x)e^{-x} + 3 \underbrace{ \int \left(e^{-x} \sin(3x) \right) }_\text{=I} \right].$$

From here, we see that the final bit in the integral is our original integral, i.e $I$, so we can substitute this in and simplify to get

$$I = - \sin(3x)e^{-x} -3 (- \cos(3x)e^{-x} + 3I)$$ $$I = - \sin(3x)e^{-x} + 3 \cos(3x)e^{-x} - 9I.$$

We can now re-arrange and solve for $I$, which is the answer to our original integral:

$$10I = - \sin(3x)e^{-x} + 3\cos(3x) e^{-x}$$ $$I = \frac{e^{-x} \left( 3 \cos(3x) - \sin(3x) \right)}{10}$$

$$\int e^{3ix}e^{-x}dx=\int e^{(3i-1)x}dx=\frac{1}{3i-1}e^{(3i-1)x}+C=-\frac{1+3i}{10}e^{-x}(\cos(3x)+i\sin(3x))+C=\frac{-\sin(3x)-3\cos(3x)}{10}e^{-x}+C$$