|
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 ?? |
|||
|
|
|
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... |
||||
|
|
