What is the integral of 9/(sqrt(1+e^{2x))} ?

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

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integral of 9/(sqrt(1+e^{2x))}

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

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

Solution steps

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

Take the constant out:

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

= 9 \cdot \int \frac{1}{1 + e ^{2 x}} d x

Apply u-substitution

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

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

Factor

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

= 9 \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

= 9 (- \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}

= 9 (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}))

Substitute back

u = 1 + e^{2 x}

= 9 (- (\frac{ln 1 + e ^{2 x} + 1}{2} - \frac{ln 1 + e ^{2 x} - 1}{2}))

Simplify

9 (- (\frac{ln 1 + e ^{2 x} + 1}{2} - \frac{ln 1 + e ^{2 x} - 1}{2})) : - 9 (\frac{1}{2} ln 1 + e^{2 x} + 1 - \frac{1}{2} ln 1 + e^{2 x} - 1)

= - 9 (\frac{1}{2} ln 1 + e^{2 x} + 1 - \frac{1}{2} ln 1 + e^{2 x} - 1)

Add a constant to the solution

= - 9 (\frac{1}{2} ln 1 + e^{2 x} + 1 - \frac{1}{2} ln 1 + e^{2 x} - 1) + C

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