What is the integral of (cos(ln(x)+1))/x ?

The integral of (cos(ln(x)+1))/x is sin(1+ln(x))+C

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integral of (cos(ln(x)+1))/x

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

sin (1 + ln (x)) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{cos ( ln ( x ) ) cos ( 1 ) - sin ( ln ( x ) ) sin ( 1 )}{x} d x

Expand

\frac{cos ( ln ( x ) ) cos ( 1 ) - sin ( ln ( x ) ) sin ( 1 )}{x} : \frac{cos ( 1 ) cos ( ln ( x ) )}{x} - \frac{sin ( 1 ) sin ( ln ( x ) )}{x}

= \int \frac{cos ( 1 ) cos ( ln ( x ) )}{x} - \frac{sin ( 1 ) sin ( ln ( x ) )}{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

= \int \frac{cos ( 1 ) cos ( ln ( x ) )}{x} d x - \int \frac{sin ( 1 ) sin ( ln ( x ) )}{x} d x

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

\int \frac{sin ( 1 ) sin ( ln ( x ) )}{x} d x = - sin (1) cos (ln (x))

= cos (1) sin (ln (x)) - (- sin (1) cos (ln (x)))

Simplify

cos (1) sin (ln (x)) - (- sin (1) cos (ln (x))) : sin (1 + ln (x))

= sin (1 + ln (x))

Add a constant to the solution

= sin (1 + ln (x)) + C

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What is the integral of (cos(ln(x)+1))/x ?
The integral of (cos(ln(x)+1))/x is sin(1+ln(x))+C