What is the integral of sinh(x)cosh(x) ?

The integral of sinh(x)cosh(x) is 1/4 cosh(2x)+C

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integral of sinh(x)cosh(x)

\int sinh (x) cosh (x) d x

\frac{1}{4} cosh (2 x) + C

Solution steps

\int sinh (x) cosh (x) d x

Rewrite using trig identities

= \int sinh (2 x) \frac{1}{2} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int sinh (2 x) d x

Apply u-substitution

: \int sinh (u) \frac{1}{2} d u

= \frac{1}{2} \cdot \int sinh (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int sinh (u) d u

Use the common integral:

\int sinh (u) d u = cosh (u)

= \frac{1}{2} \cdot \frac{1}{2} cosh (u)

Substitute back

u = 2 x

= \frac{1}{2} \cdot \frac{1}{2} cosh (2 x)

Simplify

\frac{1}{2} \cdot \frac{1}{2} cosh (2 x) : \frac{1}{4} cosh (2 x)

= \frac{1}{4} cosh (2 x)

Add a constant to the solution

= \frac{1}{4} cosh (2 x) + C

1 - x, 0

y^{^{'''}} + 3 y^{^{''}} - 4 y = 0

y^{^{''}} + 3 y^{^{'}} + 2 y = cos (2 x)

y^{^{'}} + y - e^{x} = 0

What is the integral of sinh(x)cosh(x) ?
The integral of sinh(x)cosh(x) is 1/4 cosh(2x)+C