"Need help integrating∫e8x(e16x+36)" was answered by our community

Ali Marshall

Answered question

2022-04-22

Need help integrating

\int \frac{e^{8 x}}{(e^{16 x} + 36)}

pecanjima9vo

Beginner2022-04-23Added 11 answers

So, let's try that: setting

w = e^{8 x}

, we have

d w = 8 e^{8 x}, d x

, so the integral becomes

\int \frac{e^{8 x}, d x}{{(e^{8 x})}^{2} + 36} = \frac{1}{8} \int \frac{d w}{w^{2} + 36}

Now, factor out

6^{2}

from the denominator, to get

\frac{1}{8} \int \frac{d w}{w^{2} + 36} = \frac{1}{288} \int \frac{{d w}}{{(\frac{w}{6})}^{2} + 1}

Now set

u = \frac{w}{6}

, so

6 d u = d w

. We have:

\frac{1}{288} \int \frac{d w}{{(\frac{w}{6})}^{2} + 1} = \frac{1}{48} \int \frac{d u}{u^{2} + 1}

The last integral should be immediate

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?