# Integration Question (In a loop)

• Hyari
In summary, the integration by parts method can be used to solve for the integral of e^x * sin(x). However, full equations should be written out to avoid confusion.

e^x * sin(x)

## Homework Equations

uv - integral vdu

## The Attempt at a Solution

FIRST, I set u = e^x, du = e^x, dv = sin(x)dx, v = -cos(x)

Using, uv - integral vdu

e^x * -cos(x) - [ e^x * -cos(x)dx

-e^x * cos(x) + [ e^x * cos(x)

Now I'm stuck... if I keep integrating... I'll keep going in a loop of sin/cos(x) and e^x.

You don't actually state that you are trying to integrate exsin(x) but I will assume that!

One thing that is confusing you is that you are not writing out full equations!

After your first integration by parts, you get
$$\int e^x sin(x)dx= -e^x cos(x)+\int e^x cos(x) dx$$

Now let u= ex, dv= cos(x), so that du= exdx and v= sin(x) and you get
$$\int e^x sin(x)dx= -e^x cos(x)+ e^x sin(x)- \int e^x sin(x)dx$$

Is that the "loop" you mean? Now solve that equation for
$$\int e^x sin(x)dx$$!

$$\sin x= \mathfrak{Im} \left(e^{ix}\right)$$

and the integral becomes a simple exponential.