What is the integral of (cos(ln(x)-6))/x ?

The integral of (cos(ln(x)-6))/x is sin(ln(x)-6)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (cos(ln(x)-6))/x

\int \frac{cos ( ln ( x ) - 6 )}{x} d x

sin (ln (x) - 6) + C

Solution steps

\int \frac{cos ( ln ( x ) - 6 )}{x} d x

Rewrite using trig identities

= \int \frac{cos ( ln ( x ) ) cos ( 6 ) + sin ( ln ( x ) ) sin ( 6 )}{x} d x

Expand

\frac{cos ( ln ( x ) ) cos ( 6 ) + sin ( ln ( x ) ) sin ( 6 )}{x} : \frac{cos ( 6 ) cos ( ln ( x ) )}{x} + \frac{sin ( 6 ) sin ( ln ( x ) )}{x}

= \int \frac{cos ( 6 ) cos ( ln ( x ) )}{x} + \frac{sin ( 6 ) 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 ( 6 ) cos ( ln ( x ) )}{x} d x + \int \frac{sin ( 6 ) sin ( ln ( x ) )}{x} d x

\int \frac{cos ( 6 ) cos ( ln ( x ) )}{x} d x = cos (6) sin (ln (x))

\int \frac{sin ( 6 ) sin ( ln ( x ) )}{x} d x = - sin (6) cos (ln (x))

= cos (6) sin (ln (x)) - sin (6) cos (ln (x))

Use the Angle Difference identity:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= sin (ln (x) - 6)

Add a constant to the solution

= sin (ln (x) - 6) + C

30 (2 e^{x^{2}} x^{2} + e^{x^{2}})

x \to + 4 lim (9 x + ∣ x - 4 ∣)

- sin^{3} (x)

\frac{\partial}{\partial x} (2 y sin (x) - x^{3} + ln (y))

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

What is the integral of (cos(ln(x)-6))/x ?
The integral of (cos(ln(x)-6))/x is sin(ln(x)-6)+C