What is the integral from ln(2) to ln(10) of tanh(x) ?

The integral from ln(2) to ln(10) of tanh(x) is ln(101/25)

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integral from ln(2) to ln(10) of tanh(x)

\int_{l n (2)}^{l n (10)} tanh (x) d x

ln (\frac{101}{25})

Decimal Notation

1.39624 \dots

Solution steps

\int_{l n (2)}^{l n (10)} tanh (x) d x

Rewrite using trig identities

= \int_{l n (2)}^{l n (10)} \frac{sinh ( x )}{cosh ( x )} d x

Apply u-substitution

: \int_{\frac{5}{4}}^{\frac{101}{20}} \frac{1}{u} d u

= \int_{\frac{5}{4}}^{\frac{101}{20}} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= [ln ∣ u ∣]_{\frac{5}{4}}^{\frac{101}{20}}

Compute the boundaries:

ln (\frac{101}{20}) - ln (\frac{5}{4})

= ln (\frac{101}{20}) - ln (\frac{5}{4})

Simplify

= ln (\frac{101}{25})

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What is the integral from ln(2) to ln(10) of tanh(x) ?
The integral from ln(2) to ln(10) of tanh(x) is ln(101/25)