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Consider this integral:

$$\int e^x\cdot cos(x)\space dx$$

Using integration by parts: $\int uv'\space dx=uv- \int u'v \space dx$

I have 2 possible outcomes:

1.) Using $u=e^x$, $v'=cos (x)$, $u'=e^x$ and $v=sin(x)$ gives: $$\int e^x\cdot cos(x)\space dx=e^x\cdot cos(x)-\int e^x\cdot sin(x)\space dx$$

2.) Using $u=cos (x)$, $v'=e^x (x)$, $u'=-sin(x)$ and $v=e^x$ gives:

$$\int e^x\cdot cos(x)\space dx=e^x\cdot cos(x)-\int -e^x\cdot sin(x)\space dx$$

$$\int e^x\cdot cos(x)\space dx=e^x\cdot cos(x)+\int e^x\cdot sin(x)\space dx$$

So I get:

$$e^x\cdot cos(x)-\int e^x\cdot sin(x)\space dx=e^x\cdot cos(x)+\int e^x\cdot sin(x)\space dx$$

Where have I gone wrong?

Itakura
  • 588

1 Answers1

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I wonder if there is a name for this.

We have \begin{align*} \int e^x\cos x\,dx&=e^{x}\sin x-\int e^{x}\sin x\,dx\tag 1\\ &=e^{x}\sin x-\left[-e^x\cos x-\int- e^x\cos x\,dx+C\right]\tag 2\\ &=e^x\sin x+e^x\cos x-\int e^x\cos x\,dx+C. \end{align*} where in $(1)$ I let $u = e^x, dv = \cos x$, and in $(2)$ I let $u = e^x, dv = \sin x$. Then adding $\int e^x\cos x\,dx$ to both sides and solving gives $$\int e^x\cos x\,dx = \frac{1}{2}\big[e^x\sin x+e^x\cos x+C\big].$$

Em.
  • 15,981
  • Is there a name for this? When it's $e^{-x}$ it's called a damped sinusoidal. Picture the graph of $y=\cos x$ squeezed between the graphs of $y=e^{-x}$ and $y=-e^{-x}$. – John Wayland Bales Mar 08 '16 at 05:27
  • Interesting. Yes, I see that. But I meant, when we have these "cyclic" integrals that return to itself. – Em. Mar 08 '16 at 05:29
  • The same technique can be used to integrate $e^{ax}\cos(bx)$ or $e^{ax}\sin(bx)$ to get a general formula for this type of integral. – John Wayland Bales Mar 08 '16 at 05:29
  • Yes, thank you. – Em. Mar 08 '16 at 05:31
  • I have never heard of a name for it. I always taught my students to do integration by parts by the tabular method and told them to keep going until they could integrate the bottom row of the table or until the bottom row was a multiple (other than 1) of the original integrand. – John Wayland Bales Mar 08 '16 at 05:31
  • Yes, I am aware of the tabular technique. Just like the damped sinusoidal, I wouldn't be surprised if the trait of being "cyclic" has a name. – Em. Mar 08 '16 at 05:36