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Integration by parts: help!

Prokaryotic_crap

Hi,

please help, this is driving me crazy. I'm trying to integrate:

\int e^{sx} \frac{1}{6} (e^{\frac{-x}{2}} + xe^{\frac{-x}{2}}) dx

The expression is not clear to me. You have e to the power something x. The something looks like an 8?

I can't open your WORD document because you are using docx. Can you save as a normal WORD document?

Prokaryotic_crap

OP

Original post by steve2005

The expression is not clear to me. You have e to the power something x. The something looks like an 8?

I can't open your WORD document because you are using docx. Can you save as a normal WORD document?

edit: it is e^sx

At the end of the day, you have

\frac{1}{6} \int e^{ax} + x e^{bx}\,dx

for suitable choices of a and b. Knowing your ability, this should not be difficult!

Prokaryotic_crap

OP

Original post by DFranklin

At the end of the day, you have

\frac{1}{6} \int e^{ax} + x e^{bx}\,dx

for suitable choices of a and b. Knowing your ability, this should not be difficult!

I haven't done this in a long time :P

should I multiply out the brackets first i.e. e^sx multiplied by every term in the brackets?

Yes.

Prokaryotic_crap

OP

\frac{1}{6} \int e^{\frac{sx-x}{2}} + xe^{\frac{sx-x}{2}} dx

=\frac{1}{6} [\frac{e^{\frac{sx-x}{2}}}{s-\frac{1}{2}}+[x\frac{e^{\frac{sx-x}{2}}}{s-\frac{1}{2}} - \int \frac{e^{\frac{sx-x}{2}}}{s-\frac{1}{2}} dx]

is that right so far?

(edited 13 years ago)

Looks OK. Check with Wolfram Alpha at the end...

Prokaryotic_crap

OP

Original post by Prokaryotic_crap

\frac{1}{6} \int e^{\frac{sx-x}{2}} + xe^{\frac{sx-x}{2}} dx

=\frac{1}{6} [\frac{e^{\frac{sx-x}{2}}}{s-\frac{1}{2}}+[x\frac{e^{\frac{sx-x}{2}}}{s-\frac{1}{2}} - \int \frac{e^{\frac{sx-x}{2}}}{s-\frac{1}{2}} dx]

is that right so far?

continuing...

=\frac{1}{6} [\frac{e^{\frac{sx-x}{2}}}{s-\frac{1}{2}} + [x\frac{e^{\frac{sx-x}{2}}}{s-\frac{1}{2}} - [\frac{1}{s-\frac{1}{2}} \int e^{\frac{sx-x}{2}} dx]

\frac{1}{6} [\frac{e^{\frac{sx-x}{2}}}{s-\frac{1}{2}} + [x\frac{e^{\frac{sx-x}{2}}}{s-\frac{1}{2}} - [\frac{1}{s-\frac{1}{2}} [\frac{e^{\frac{sx-x}{2}}}{s-\frac{1}{2}}]]]]

still ok?

(edited 13 years ago)

Looks about right. You would make life much simpler if you defined

a = s - \frac{1}{2}

, then you'd just be integrating

\frac{1}{6} \int e^{ax} + x e^{ax}\,dx

Prokaryotic_crap

OP

Original post by DFranklin

Looks about right. You would make life much simpler if you defined

a = s - \frac{1}{2}

, then you'd just be integrating

\frac{1}{6} \int e^{ax} + x e^{ax}\,dx

how can I proceed from the last line of my working?

Original post by Prokaryotic_crap

how can I proceed from the last line of my working?

You've already done the integration fine, any tidying up is just algebra.

If you define

a= s-\frac{1}{2}

as DFranklin said, you can end up with

\displaystyle \frac{e^{ax}}{6a}\left(1+x-\frac{1}{a}\right)

(edited 13 years ago)

Prokaryotic_crap

OP

Original post by .matt

You've already done the integration fine, any tidying up is just algebra.

If you define

a= s-\frac{1}{2}

as DFranklin said, you can end up with

\displaystyle \frac{e^{ax}}{6a}\left(1+x-\frac{1}{a}\right)

how canI tidy it up further?

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