What is the integral of (sin^2(ln(x)))/(xcos(ln(x))) ?

The integral of (sin^2(ln(x)))/(xcos(ln(x))) is ln|tan(ln(x))+sec(ln(x))|-sin(ln(x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (sin^2(ln(x)))/(xcos(ln(x)))

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

ln ∣ tan (ln (x)) + sec (ln (x)) ∣ - sin (ln (x)) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1 - cos ^{2} ( ln ( x ) )}{x cos ( ln ( x ) )} d x

Expand

\frac{1 - cos ^{2} ( ln ( x ) )}{x cos ( ln ( x ) )} : \frac{1}{x cos ( ln ( x ) )} - \frac{cos ( ln ( x ) )}{x}

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

\int \frac{1}{x cos ( ln ( x ) )} d x = ln ∣ tan (ln (x)) + sec (ln (x)) ∣

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

= ln ∣ tan (ln (x)) + sec (ln (x)) ∣ - sin (ln (x))

Add a constant to the solution

= ln ∣ tan (ln (x)) + sec (ln (x)) ∣ - sin (ln (x)) + C

What is the integral of (sin^2(ln(x)))/(xcos(ln(x))) ?
The integral of (sin^2(ln(x)))/(xcos(ln(x))) is ln|tan(ln(x))+sec(ln(x))|-sin(ln(x))+C