What is the integral of sec(t)[tan(t)+sec(t)] ?

The integral of sec(t)[tan(t)+sec(t)] is sec(t)+tan(t)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sec(t)[tan(t)+sec(t)]

\int sec (t) [tan (t) + sec (t)] d t

sec (t) + tan (t) + C

Solution steps

\int sec (t) [tan (t) + sec (t)] d t

Rewrite using trig identities

= \int \frac{sin ( t ) + 1}{cos ^{2} ( t )} d t

Expand

\frac{sin ( t ) + 1}{cos ^{2} ( t )} : \frac{sin ( t )}{cos ^{2} ( t )} + \frac{1}{cos ^{2} ( t )}

= \int \frac{sin ( t )}{cos ^{2} ( t )} + \frac{1}{cos ^{2} ( t )} d t

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{sin ( t )}{cos ^{2} ( t )} d t + \int \frac{1}{cos ^{2} ( t )} d t

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

\int \frac{1}{cos ^{2} ( t )} d t = tan (t)

= sec (t) + tan (t)

Add a constant to the solution

= sec (t) + tan (t) + C

\int \frac{1}{( x ^{2} - 2 x - 3 ) ^{\frac{3}{2}}} d x

\int \frac{1}{4 x - 3 a} d x

\int (8 x + 9) 4 x^{2} + 9 x d x

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

\int 7 x^{8} d x

What is the integral of sec(t)[tan(t)+sec(t)] ?
The integral of sec(t)[tan(t)+sec(t)] is sec(t)+tan(t)+C