What is the integral of (sinh(x))/(cosh^3(x)) ?

The integral of (sinh(x))/(cosh^3(x)) is -1/2 sech^2(x)+C

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integral of (sinh(x))/(cosh^3(x))

\int \frac{sinh ( x )}{cosh ^{3} ( x )} d x

- \frac{1}{2} sech^{2} (x) + C

Solution steps

\int \frac{sinh ( x )}{cosh ^{3} ( x )} d x

Rewrite using trig identities

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

Use the Hyperbolic identity:

\frac{1}{cosh ( x )} = sech (x)

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

Apply u-substitution

: \int - \frac{1}{2} d u

= \int - \frac{1}{2} d u

Integral of a constant:

\int a d x = a x

= (- \frac{1}{2}) u

Substitute back

u = sech^{2} (x)

= (- \frac{1}{2}) sech^{2} (x)

Simplify

= - \frac{1}{2} sech^{2} (x)

Add a constant to the solution

= - \frac{1}{2} sech^{2} (x) + C

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What is the integral of (sinh(x))/(cosh^3(x)) ?
The integral of (sinh(x))/(cosh^3(x)) is -1/2 sech^2(x)+C