What is the integral from 0 to ln(20) of tanh(x) ?

The integral from 0 to ln(20) of tanh(x) is ln(401/40)

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integral from 0 to ln(20) of tanh(x)

\int_{0}^{l n (20)} tanh (x) d x

ln (\frac{401}{40})

Decimal Notation

2.30508 \dots

Solution steps

\int_{0}^{l n (20)} tanh (x) d x

Rewrite using trig identities

= \int_{0}^{l n (20)} \frac{sinh ( x )}{cosh ( x )} d x

Apply u-substitution

: \int_{1}^{\frac{401}{40}} \frac{1}{u} d u

= \int_{1}^{\frac{401}{40}} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= [ln ∣ u ∣]_{1}^{\frac{401}{40}}

Compute the boundaries:

ln (\frac{401}{40})

= ln (\frac{401}{40})

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What is the integral from 0 to ln(20) of tanh(x) ?
The integral from 0 to ln(20) of tanh(x) is ln(401/40)