What is the integral of 1/(tan(θ)sec(θ)) ?

The integral of 1/(tan(θ)sec(θ)) is ln|tan(θ/2)|+cos(θ)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(tan(θ)sec(θ))

\int \frac{1}{tan ( θ ) sec ( θ )} d θ

ln tan (\frac{θ}{2}) + cos (θ) + C

Solution steps

\int \frac{1}{tan ( θ ) sec ( θ )} d θ

Rewrite using trig identities

= \int \frac{1 - sin ^{2} ( θ )}{sin ( θ )} d θ

Expand

\frac{1 - sin ^{2} ( θ )}{sin ( θ )} : \frac{1}{sin ( θ )} - sin (θ)

= \int \frac{1}{sin ( θ )} - sin (θ) d θ

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}{sin ( θ )} d θ - \int sin (θ) d θ

\int \frac{1}{sin ( θ )} d θ = ln tan (\frac{θ}{2})

\int sin (θ) d θ = - cos (θ)

= ln tan (\frac{θ}{2}) - (- cos (θ))

Simplify

= ln tan (\frac{θ}{2}) + cos (θ)

Add a constant to the solution

= ln tan (\frac{θ}{2}) + cos (θ) + C

\int 5 x^{3} e^{(5 x - 3)} d x

\int \frac{1}{x ^{2} + 4 ^{2}} d x

\int (6 x e^{x^{2} + 3}) d x

\int 5 (5 x + 12)^{3} d x

\int (e^{4 x} + 3 x^{\frac{1}{4}}) d x

What is the integral of 1/(tan(θ)sec(θ)) ?
The integral of 1/(tan(θ)sec(θ)) is ln|tan(θ/2)|+cos(θ)+C