What is the integral of xtanh^2(x) ?

The integral of xtanh^2(x) is x^2-xtanh(x)-(x^2)/2+ln|cosh(x)|+C

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integral of xtanh^2(x)

\int x tanh^{2} (x) d x

x^{2} - x tanh (x) - \frac{x ^{2}}{2} + ln ∣ cosh (x) ∣ + C

Solution steps

\int x tanh^{2} (x) d x

Apply Integration By Parts

: x (x - tanh (x)) - \int x - tanh (x) d x

= x (x - tanh (x)) - \int x - tanh (x) d x

\int x - tanh (x) d x = \frac{x ^{2}}{2} - ln ∣ cosh (x) ∣

= x (x - tanh (x)) - (\frac{x ^{2}}{2} - ln ∣ cosh (x) ∣)

Simplify

x (x - tanh (x)) - (\frac{x ^{2}}{2} - ln ∣ cosh (x) ∣) : x^{2} - x tanh (x) - \frac{x ^{2}}{2} + ln ∣ cosh (x) ∣

= x^{2} - x tanh (x) - \frac{x ^{2}}{2} + ln ∣ cosh (x) ∣

Add a constant to the solution

= x^{2} - x tanh (x) - \frac{x ^{2}}{2} + ln ∣ cosh (x) ∣ + C

e^{- \frac{1}{x}} + 4 x

f (x) = - \frac{x sin ( \frac{x ^{2}}{4} )}{2}

\int e^{x} 11 + e^{x} d x

104 - 80 e^{- 0.2 t}

arcsin (2 x + 1)

What is the integral of xtanh^2(x) ?
The integral of xtanh^2(x) is x^2-xtanh(x)-(x^2)/2+ln|cosh(x)|+C