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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Apply Trigonometric Substitution

= 2 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 2 \cdot 1 \cdot v

Substitute back

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

Simplify

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

Add a constant to the solution

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

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