What is the integral of (e^{2x})/(sqrt(9-e^{4x))} ?

The integral of (e^{2x})/(sqrt(9-e^{4x))} is 1/2 arcsin(1/3 e^{2x})+C

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integral of (e^{2x})/(sqrt(9-e^{4x))}

\int \frac{e ^{2 x}}{9 - e ^{4 x}} d x

\frac{1}{2} arcsin (\frac{1}{3} e^{2 x}) + C

Solution steps

\int \frac{e ^{2 x}}{9 - e ^{4 x}} d x

Apply u-substitution

: \int \frac{1}{2 9 - u ^{2}} d u

= \int \frac{1}{2 9 - u ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{9 - u ^{2}} d u

Apply Trigonometric Substitution

: \int 1 d v

= \frac{1}{2} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{2} \cdot 1 \cdot v

Substitute back

= \frac{1}{2} \cdot 1 \cdot arcsin (\frac{1}{3} e^{2 x})

Simplify

= \frac{1}{2} arcsin (\frac{1}{3} e^{2 x})

Add a constant to the solution

= \frac{1}{2} arcsin (\frac{1}{3} e^{2 x}) + C

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What is the integral of (e^{2x})/(sqrt(9-e^{4x))} ?
The integral of (e^{2x})/(sqrt(9-e^{4x))} is 1/2 arcsin(1/3 e^{2x})+C