What is the integral of (3e^x)/(sqrt(4-e^{2x))} ?

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

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integral of (3e^x)/(sqrt(4-e^{2x))}

\int \frac{3 e ^{x}}{4 - e ^{2 x}} d x

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

Solution steps

\int \frac{3 e ^{x}}{4 - e ^{2 x}} d x

Take the constant out:

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

= 3 \cdot \int \frac{e ^{x}}{4 - e ^{2 x}} d x

Apply u-substitution

: \int \frac{1}{4 - u ^{2}} d u

= 3 \cdot \int \frac{1}{4 - u ^{2}} d u

Apply Trigonometric Substitution

: \int 1 d v

= 3 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 3 \cdot 1 \cdot v

Substitute back

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

Simplify

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

Add a constant to the solution

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

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