What is the integral of (2cosh(sqrt(x)))/(sqrt(x)) ?

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

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integral of (2cosh(sqrt(x)))/(sqrt(x))

\int \frac{2 cosh ( x )}{x} d x

4 sinh (x) + C

Solution steps

\int \frac{2 cosh ( x )}{x} d x

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \int 2 cosh (u) d u

Take the constant out:

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

= 2 \cdot 2 \cdot \int cosh (u) d u

Use the common integral:

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

= 2 \cdot 2 sinh (u)

Substitute back

u = x

= 2 \cdot 2 sinh (x)

Simplify

= 4 sinh (x)

Add a constant to the solution

= 4 sinh (x) + C

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