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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Apply Trigonometric Substitution

: \int 1 d v

= \int 1 d v

Integral of a constant:

\int a d x = a x

= 1 \cdot v

Substitute back

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

Simplify

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

Add a constant to the solution

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

\frac{\partial}{\partial u} (u^{5} + v^{7})

\frac{\partial}{\partial x} (y^{2} + 16)

\frac{d}{d x} (\frac{x + 1}{x + 5})

y^{^{''}} - 8 y^{^{'}} + 25 y = 208 sin (3 x)

y^{^{'}} = \frac{2 x y}{x ^{2} - y ^{2}}

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