dx-a-2-be-cx-

Question Number 187530 by sciencestudentW last updated on 18/Feb/23

\int \frac{d x}{\sqrt{a^{2} + {b e}^{c x}}} = ?

Answered by anurup last updated on 18/Feb/23

a^{2} + {b e}^{c x} = t^{2}

\Rightarrow {b c e}^{c x} d x = 2 t d t

\Rightarrow d x = \frac{2 t d t}{c (t^{2} - a^{2})}

\int \frac{2 t d t}{c (t^{2} - a^{2}) t}

= \frac{2}{c} \int \frac{d t}{t^{2} - a^{2}}

= \frac{1}{c a} \ln ∣ \frac{t - a}{t + a} ∣ + k

= \frac{1}{c a} \ln ∣ \frac{\sqrt{a^{2} + {b e}^{c x}} - a}{\sqrt{a^{2} + {b e}^{c x}} + a} ∣ + k

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