What is the integral of (4cosh(5x))/(sinh^3(5x)) ?

The integral of (4cosh(5x))/(sinh^3(5x)) is -2/5 csch^2(5x)+C

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integral of (4cosh(5x))/(sinh^3(5x))

\int \frac{4 cosh ( 5 x )}{sinh ^{3} ( 5 x )} d x

- \frac{2}{5} csch^{2} (5 x) + C

Solution steps

\int \frac{4 cosh ( 5 x )}{sinh ^{3} ( 5 x )} d x

Take the constant out:

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

= 4 \cdot \int \frac{cosh ( 5 x )}{sinh ^{3} ( 5 x )} d x

Apply u-substitution

: \int \frac{1}{15 u ^{\frac{5}{3}}} d u

= 4 \cdot \int \frac{1}{15 u ^{\frac{5}{3}}} d u

Take the constant out:

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

= 4 \cdot \frac{1}{15} \cdot \int \frac{1}{u ^{\frac{5}{3}}} d u

Apply the Power Rule

: - \frac{3}{2 u ^{\frac{2}{3}}}

= 4 \cdot \frac{1}{15} (- \frac{3}{2 u ^{\frac{2}{3}}})

Substitute back

u = sinh^{3} (5 x)

= 4 \cdot \frac{1}{15} - \frac{3}{2 ( sinh ^{3} ( 5 x ) ) ^{\frac{2}{3}}}

Simplify

4 \cdot \frac{1}{15} - \frac{3}{2 ( sinh ^{3} ( 5 x ) ) ^{\frac{2}{3}}} : - \frac{2}{5} csch^{2} (5 x)

= - \frac{2}{5} csch^{2} (5 x)

Add a constant to the solution

= - \frac{2}{5} csch^{2} (5 x) + C

f (x) = 3 x^{2} - 4 x + 5

\frac{d}{d x} (ln (tanh (2 x)))

y^{^{''}} - 3 y^{^{'}} - 4 y = 10 e^{- x} + 4 x + 3

\frac{d}{d x} ((5 x - 3) (2 x + 5)^{2})

\frac{2 x}{9 - tan ( x )}

What is the integral of (4cosh(5x))/(sinh^3(5x)) ?
The integral of (4cosh(5x))/(sinh^3(5x)) is -2/5 csch^2(5x)+C