What is the integral of cosh(2ln(x)) ?

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

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integral of cosh(2ln(x))

\int cosh (2 ln (x)) d x

\frac{1}{2} (- \frac{1}{x} + \frac{x ^{3}}{3}) + C

Solution steps

\int cosh (2 ln (x)) d x

Rewrite using trig identities

= \int \frac{1 + x ^{4}}{2 x ^{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 \frac{1 + x ^{4}}{x ^{2}} d x

Expand

\frac{1 + x ^{4}}{x ^{2}} : \frac{1}{x ^{2}} + x^{2}

= \frac{1}{2} \cdot \int \frac{1}{x ^{2}} + x^{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{x ^{2}} d x = - \frac{1}{x}

\int x^{2} d x = \frac{x ^{3}}{3}

= \frac{1}{2} (- \frac{1}{x} + \frac{x ^{3}}{3})

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{x} + \frac{x ^{3}}{3}) + C

\int \frac{sin ( θ )}{cos ^{2} ( θ )} d θ

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

\frac{d}{d x} (\frac{9 x}{ln ( x )})

f (x) = \frac{x + 1}{x}

(x - \frac{6}{x})^{8}

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