What is the integral of tan(x)(6cos(x)-6sec(x)) ?

The integral of tan(x)(6cos(x)-6sec(x)) is -6cos(x)-6sec(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)(6cos(x)-6sec(x))

\int tan (x) (6 cos (x) - 6 sec (x)) d x

- 6 cos (x) - 6 sec (x) + C

Solution steps

\int tan (x) (6 cos (x) - 6 sec (x)) d x

Rewrite using trig identities

= \int \frac{sin ( x ) ( 6 cos ^{2} ( x ) - 6 )}{cos ^{2} ( x )} d x

Expand

\frac{sin ( x ) ( 6 cos ^{2} ( x ) - 6 )}{cos ^{2} ( x )} : 6 sin (x) - \frac{6 sin ( x )}{cos ^{2} ( x )}

= \int 6 sin (x) - \frac{6 sin ( x )}{cos ^{2} ( 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 6 sin (x) d x - \int \frac{6 sin ( x )}{cos ^{2} ( x )} d x

\int 6 sin (x) d x = - 6 cos (x)

\int \frac{6 sin ( x )}{cos ^{2} ( x )} d x = 6 sec (x)

= - 6 cos (x) - 6 sec (x)

Add a constant to the solution

= - 6 cos (x) - 6 sec (x) + C

What is the integral of tan(x)(6cos(x)-6sec(x)) ?
The integral of tan(x)(6cos(x)-6sec(x)) is -6cos(x)-6sec(x)+C