What is the integral of 3/(sqrt(1+e^{2x))} ?

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

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integral of 3/(sqrt(1+e^{2x))}

\int \frac{3}{1 + e ^{2 x}} d x

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

Solution steps

\int \frac{3}{1 + 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{1}{1 + e ^{2 x}} d x

Apply u-substitution

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

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

Factor

u^{2} - 1 : - (- u^{2} + 1)

= 3 \cdot \int \frac{1}{- ( - u ^{2} + 1 )} d u

Take the constant out:

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

= 3 (- \int \frac{1}{- u ^{2} + 1} d u)

Use the common integral:

\int \frac{1}{- u ^{2} + 1} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= 3 (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}))

Substitute back

u = 1 + e^{2 x}

= 3 (- (\frac{ln 1 + e ^{2 x} + 1}{2} - \frac{ln 1 + e ^{2 x} - 1}{2}))

Simplify

3 (- (\frac{ln 1 + e ^{2 x} + 1}{2} - \frac{ln 1 + e ^{2 x} - 1}{2})) : - 3 (\frac{1}{2} ln 1 + e^{2 x} + 1 - \frac{1}{2} ln 1 + e^{2 x} - 1)

= - 3 (\frac{1}{2} ln 1 + e^{2 x} + 1 - \frac{1}{2} ln 1 + e^{2 x} - 1)

Add a constant to the solution

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

\int_{0}^{1} x^{7} (1 + x^{8})^{7} d x

\int \frac{x + 9}{1 - 2 x - 2 x ^{2}} d x

\int_{0}^{1} \frac{y}{e ^{9 y}} d y

10 (t + 1) y^{^{'}} - 8 y = 16 t

y = 2 + 4 x^{2} - 2 x^{3}

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