What is the integral from 0 to ln(5) of tanh(x) ?

The integral from 0 to ln(5) of tanh(x) is ln(13/5)

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integral from 0 to ln(5) of tanh(x)

\int_{0}^{l n (5)} tanh (x) d x

ln (\frac{13}{5})

Decimal

0.95551 \dots

Solution steps

\int_{0}^{l n (5)} tanh (x) d x

Rewrite using trig identities

= \int_{0}^{l n (5)} \frac{sinh ( x )}{cosh ( x )} d x

Apply u-substitution

: \int_{1}^{\frac{13}{5}} \frac{1}{u} d u

= \int_{1}^{\frac{13}{5}} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= [ln ∣ u ∣]_{1}^{\frac{13}{5}}

Compute the boundaries:

ln (\frac{13}{5})

= ln (\frac{13}{5})

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What is the integral from 0 to ln(5) of tanh(x) ?
The integral from 0 to ln(5) of tanh(x) is ln(13/5)