What is the integral of sec(t)(3sec(t)+4tan(t)) ?

The integral of sec(t)(3sec(t)+4tan(t)) is 3tan(t)+4sec(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)(3sec(t)+4tan(t))

\int sec (t) (3 sec (t) + 4 tan (t)) d t

3 tan (t) + 4 sec (t) + C

Solution steps

\int sec (t) (3 sec (t) + 4 tan (t)) d t

Rewrite using trig identities

= \int \frac{3 + 4 sin ( t )}{cos ^{2} ( t )} d t

Expand

\frac{3 + 4 sin ( t )}{cos ^{2} ( t )} : \frac{3}{cos ^{2} ( t )} + \frac{4 sin ( t )}{cos ^{2} ( t )}

= \int \frac{3}{cos ^{2} ( t )} + \frac{4 sin ( t )}{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{3}{cos ^{2} ( t )} d t + \int \frac{4 sin ( t )}{cos ^{2} ( t )} d t

\int \frac{3}{cos ^{2} ( t )} d t = 3 tan (t)

\int \frac{4 sin ( t )}{cos ^{2} ( t )} d t = 4 sec (t)

= 3 tan (t) + 4 sec (t)

Add a constant to the solution

= 3 tan (t) + 4 sec (t) + C

\int \frac{x - 3}{- x ^{2} + 5 x - 6} d x

x sin (x)

\int 8 (x^{7} + 7^{x} + 7 x + 7) d x

\int \frac{5 x}{x ^{2} + 6} d x

\int x^{2} x^{3} - 6 d x

What is the integral of sec(t)(3sec(t)+4tan(t)) ?
The integral of sec(t)(3sec(t)+4tan(t)) is 3tan(t)+4sec(t)+C