What is the integral of (tan(x)sin(x))/1 ?

The integral of (tan(x)sin(x))/1 is ln|tan(x)+sec(x)|-sin(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (tan(x)sin(x))/1

\int \frac{tan ( x ) sin ( x )}{1} d x

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

Solution steps

\int \frac{tan ( x ) sin ( x )}{1} d x

Simplify

= \int tan (x) sin (x) d x

Rewrite using trig identities

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

Expand

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

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

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

\int cos (x) d x = sin (x)

= ln ∣ tan (x) + sec (x) ∣ - sin (x)

Add a constant to the solution

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

What is the integral of (tan(x)sin(x))/1 ?
The integral of (tan(x)sin(x))/1 is ln|tan(x)+sec(x)|-sin(x)+C