What is the integral of cos^2(ln(x)) ?

The integral of cos^2(ln(x)) is 1/10 (5x+2xsin(2ln(x))+xcos(2ln(x)))+C

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integral of cos^2(ln(x))

\int cos^{2} (ln (x)) d x

\frac{1}{10} (5 x + 2 x sin (2 ln (x)) + x cos (2 ln (x))) + C

Solution steps

\int cos^{2} (ln (x)) d x

Rewrite using trig identities

= \int \frac{1 + cos ( 2 ln ( 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 1 + cos (2 ln (x)) 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 1 d x + \int cos (2 ln (x)) d x)

\int 1 d x = x

\int cos (2 ln (x)) d x = \frac{2}{5} x sin (2 ln (x)) + \frac{1}{5} x cos (2 ln (x))

= \frac{1}{2} (x + \frac{2}{5} x sin (2 ln (x)) + \frac{1}{5} x cos (2 ln (x)))

Simplify

\frac{1}{2} (x + \frac{2}{5} x sin (2 ln (x)) + \frac{1}{5} x cos (2 ln (x))) : \frac{1}{10} (5 x + 2 x sin (2 ln (x)) + x cos (2 ln (x)))

= \frac{1}{10} (5 x + 2 x sin (2 ln (x)) + x cos (2 ln (x)))

Add a constant to the solution

= \frac{1}{10} (5 x + 2 x sin (2 ln (x)) + x cos (2 ln (x))) + C

\frac{\partial}{\partial x} (y + (\frac{1}{\frac{1}{x} + \frac{1}{z}}))

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

n \to \infty lim (∣ n ∣)

\int e^{\frac{4}{x}} d x

ln (2 x - x^{2})

What is the integral of cos^2(ln(x)) ?
The integral of cos^2(ln(x)) is 1/10 (5x+2xsin(2ln(x))+xcos(2ln(x)))+C