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

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

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integral of cos(ln(x))

\int cos (ln (x)) d x

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

Solution steps

\int cos (ln (x)) d x

Apply u-substitution

: \int cos (u) e^{u} d u

= \int cos (u) e^{u} d u

Apply Integration By Parts

: e^{u} sin (u) - \int e^{u} sin (u) d u

= e^{u} sin (u) - \int e^{u} sin (u) d u

Apply Integration By Parts

: - e^{u} cos (u) - \int - e^{u} cos (u) d u

= e^{u} sin (u) - (- e^{u} cos (u) - \int - e^{u} cos (u) d u)

Take the constant out:

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

= e^{u} sin (u) - (- e^{u} cos (u) - (- \int e^{u} cos (u) d u))

Therefore

\int e^{u} cos (u) d u = e^{u} sin (u) - (- e^{u} cos (u) - (- \int e^{u} cos (u) d u))

Isolate

\int e^{u} cos (u) d u

= \frac{e ^{u} sin ( u )}{2} + \frac{e ^{u} cos ( u )}{2}

Substitute back

u = ln (x)

= \frac{e ^{l n (x)} sin ( ln ( x ) )}{2} + \frac{e ^{l n (x)} cos ( ln ( x ) )}{2}

Simplify

\frac{e ^{l n (x)} sin ( ln ( x ) )}{2} + \frac{e ^{l n (x)} cos ( ln ( x ) )}{2} : \frac{1}{2} x sin (ln (x)) + \frac{1}{2} x cos (ln (x))

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

Add a constant to the solution

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

\int \frac{x ^{2}}{49 + x ^{2}} d x

\int x (1 + x^{3}) d x

y^{^{'}} + y = 2, y (0) = 0

(- 5.3) (7.9)

\int_{0}^{1} 5 x e^{- x^{2}} d x

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