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i am stuck with this integral

$$ \int_{0}^{a}\sqrt{a-Be^{cx}} $$

here a,B and c are real numbers, this is equivalent to

$$ \frac{1}{c} \int_{1}^{e^{ca}} \sqrt{a-Bu} \frac{du}{u} $$

how could i evaluate this ? i have looked tables but can not find an answer what is the next step ??

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1 Answer

up vote 4 down vote accepted

Why didn't you rather choose the substitution

$$u=Be^{cx}\Longrightarrow du=Bce^{cx}dx\Longrightarrow dx=\frac{du}{uc}\Longrightarrow$$

$$\int\limits_0^a\sqrt{a-Be^{cx}}\,dx=\frac{1}{c}\int\limits_B^{Be^{ca}}\frac{\sqrt{a-u}}{u}du$$

and now make the substitution

$$y^2=a-u\Longrightarrow 2ydy=-du\Longrightarrow \frac{1}{c}\int\limits_B^{Be^{ca}}\frac{\sqrt{a-u}}{u}du=\frac{2}{c}\int\limits_{\sqrt{a-Be^{ca}}}^{\sqrt{a-B}}\frac{y^2}{a-y^2}dy$$

which already is an almost automatic integral...

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