What is the integral of sqrt(tanh(x))sech^2(x) ?

The integral of sqrt(tanh(x))sech^2(x) is 2/3 tanh^{3/2}(x)+C

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integral of sqrt(tanh(x))sech^2(x)

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

\frac{2}{3} tanh^{\frac{3}{2}} (x) + C

Solution steps

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

Apply u-substitution

= \int 2 u^{2} d u

Take the constant out:

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

= 2 \cdot \int u^{2} d u

Apply the Power Rule

= 2 \cdot \frac{u ^{3}}{3}

Substitute back

u = tanh (x)

= 2 \cdot \frac{( tanh ( x ) ) ^{3}}{3}

Simplify

2 \cdot \frac{( tanh ( x ) ) ^{3}}{3} : \frac{2}{3} tanh^{\frac{3}{2}} (x)

= \frac{2}{3} tanh^{\frac{3}{2}} (x)

Add a constant to the solution

= \frac{2}{3} tanh^{\frac{3}{2}} (x) + C

What is the integral of sqrt(tanh(x))sech^2(x) ?
The integral of sqrt(tanh(x))sech^2(x) is 2/3 tanh^{3/2}(x)+C