# Thread: Integration By Parts...e^x * cosx dx

1. ## Integration By Parts...e^x * cosx dx

Let INT = integral symbol

INT( e^x)(cosx) dx

What is u and dv here to get me started?

2. ## Re: Integration By Parts...e^x * cosx dx

use ILATE LAW
if u=first fn(higher precedence) which here is of cos(x)
v=second fn IS e^(x)
now
I=COS(X)INTEGRATE(E^(X))-INTEGRATE(DIFF(COS(X)INTEGRATE(E^(X)))
I=COS(X)E^(X)+INTEGRATE(SIN(X)E^(X))
NOW APPLY INTEGRATION OF PARTS ONE MORE TIME
I=COS(X)E^(X)+E^(X)SIN(X)-I WHERE I=INTEGRATE(E^(X)COS(X)) [UNDERSTAND IN TERMS OF RECURSION]
SOLVING I U GET
I=E^(X)(COS(X)+SIN(X))/2+CONSTANT

3. ## Re: Integration By Parts...e^x * cosx dx

Originally Posted by prasum
use ILATE LAW
if u=first fn(higher precedence) which here is of cos(x)
v=second fn IS e^(x)
now
I=COS(X)INTEGRATE(E^(X))-INTEGRATE(DIFF(COS(X)INTEGRATE(E^(X)))
I=COS(X)E^(X)+INTEGRATE(SIN(X)E^(X))
NOW APPLY INTEGRATION OF PARTS ONE MORE TIME
I=COS(X)E^(X)+E^(X)SIN(X)-I WHERE I=INTEGRATE(E^(X)COS(X)) [UNDERSTAND IN TERMS OF RECURSION]
SOLVING I U GET
I=E^(X)(COS(X)+SIN(X))/2+CONSTANT

4. ## Re: Integration By Parts...e^x * cosx dx

Originally Posted by nycmath
Let INT = integral symbol

INT( e^x)(cosx) dx

What is u and dv here to get me started?
Either way will work:
a) Let $u= e^x$ and $dv= cos(x) dx$. Then $du= e^x dx$ and $v= sin(x)$ so we have $\int e^x cos(x)dx= uv- \int v du= (e^x sin(x))- \int e^x sin(x) dx$. To integrate $\int e^x sin(x)$, let $u= e^x$ and $dv= sin(x)dx$. Then $du= e^x dx$ and $v= -cos(x)$ so we have $\int e^x sin(x)dx= uv- \int vdu= -e^x cos(x)+ \int e^x cos(x)dx$.

Putting those together,
$\int e^x cos(x)dx= e^x sin(x)- (-e^x cos(x)+ e^x cos(x)dx)= e^x(sin(x)+ cos(x))- \int e^x cos(x)dx$
and then $2\int e^x cos(x)dx= e^x(sin(x)+ cos(x))$,
$\int e^x cos(x)dx= \frac{1}{2}e^{x}(sin(x)+ cos(x))+ C$.

Now, you try it letting $u= cos(x)$, $dv= e^x dx$.

(By the way- when you are not sure which of two functions to take as "u" and which to take as "dv", try both!)

5. ## Re: Integration By Parts...e^x * cosx dx

You don't use LaTeX, but you want prasum to?

- Hollywood

6. ## Re: Integration By Parts...e^x * cosx dx

Originally Posted by hollywood
You don't use LaTeX, but you want prasum to?

- Hollywood
I never learned how to use LaTeX.

7. ## Re: Integration By Parts...e^x * cosx dx

Originally Posted by nycmath
I never learned how to use LaTeX.
You should never ask someone to do something you are not willing to do yourself. We have a LaTeX subforum with tutorials, not to mention the world at our fingertips with Google, so learn it!

8. ## Re: Integration By Parts...e^x * cosx dx

Originally Posted by Prove It
You should never ask someone to do something you are not willing to do yourself. We have a LaTeX subforum with tutorials, not to mention the world at our fingertips with Google, so learn it!

Right now my concern is calculus.

9. ## Re: Integration By Parts...e^x * cosx dx

You seem to be missing the point- whatever excuse (reason) you have for not using Latex, prasum, or any other person here, may have the same reason. Who are to ask that other people do what you, for whatever reason, do not do?

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