Evaluate: $\int \frac{e^{x}}{\sqrt{5 - 4 e^{x} - e^{2 x}}} dx$

Solution

$I = \int \frac{e^{x}}{\sqrt{5 - 4 e^{x} - e^{2 x}}} dx Let e^{x} = t \Rightarrow e^{x} dx = dt$

Now integral I becomes,

$I = \int \frac{dt}{\sqrt{5 - 4 t - t^{2}}} \Rightarrow I = \int \frac{dt}{\sqrt{5 + 4 - 4 - 4 t - t^{2}}} \Rightarrow I = \int \frac{dt}{\sqrt{9 - (4 + 4 t + t^{2})}} \Rightarrow I = \int \frac{dt}{\sqrt{9 - (t + 2)^{2}}} \Rightarrow I = \int \frac{dt}{\sqrt{3^{2} - (t + 2)^{2}}} \Rightarrow I = \sin^{- 1} \frac{(t + 2)}{3} + C \Rightarrow I = \sin^{- 1} \frac{(e^{x} + 2)}{3} + C$

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Integrals

Evaluate: $\int \frac{e^{x}}{\sqrt{5 - 4 e^{x} - e^{2 x}}} dx$

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Integrals

Evaluate: ∫ ex5 - 4 ex - e2x dx

Some More Questions From Integrals Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Evaluate: $\int \frac{e^{x}}{\sqrt{5 - 4 e^{x} - e^{2 x}}} dx$