What is the integral of (3^x)/(1+3^x) ?

The integral of (3^x)/(1+3^x) is 1/(ln(3))ln|1+3^x|+C

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integral of (3^x)/(1+3^x)

\int \frac{3 ^{x}}{1 + 3 ^{x}} d x

\frac{1}{ln ( 3 )} ln ∣ 1 + 3^{x} ∣ + C

Solution steps

\int \frac{3 ^{x}}{1 + 3 ^{x}} d x

Apply u-substitution

: \int \frac{1}{ln ( 3 ) u} d u

= \int \frac{1}{ln ( 3 ) u} d u

Take the constant out:

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

= \frac{1}{ln ( 3 )} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{ln ( 3 )} ln ∣ u ∣

Substitute back

u = 1 + 3^{x}

= \frac{1}{ln ( 3 )} ln ∣ 1 + 3^{x} ∣

Add a constant to the solution

= \frac{1}{ln ( 3 )} ln ∣ 1 + 3^{x} ∣ + C

f (x) = 3 x^{2} + 5 x, at x = - 2

x \to \frac{π}{2} lim (\frac{x}{cos ( x )})

\frac{d y}{d x}, x + 4 y = 1

y^{^{''}} + 7 y^{^{'}} + 12 y = 0, y (0) = 1, y^{^{'}} (0) = 0

\int 18 sin^{3} (x co) s^{2} x

What is the integral of (3^x)/(1+3^x) ?
The integral of (3^x)/(1+3^x) is 1/(ln(3))ln|1+3^x|+C