What is the integral of sec(t)(8sec(t)+7tan(t)) ?

The integral of sec(t)(8sec(t)+7tan(t)) is 8tan(t)+7sec(t)+C

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integral of sec(t)(8sec(t)+7tan(t))

\int sec (t) (8 sec (t) + 7 tan (t)) d t

8 tan (t) + 7 sec (t) + C

Solution steps

\int sec (t) (8 sec (t) + 7 tan (t)) d t

Rewrite using trig identities

= \int \frac{8 + 7 sin ( t )}{cos ^{2} ( t )} d t

Expand

\frac{8 + 7 sin ( t )}{cos ^{2} ( t )} : \frac{8}{cos ^{2} ( t )} + \frac{7 sin ( t )}{cos ^{2} ( t )}

= \int \frac{8}{cos ^{2} ( t )} + \frac{7 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{8}{cos ^{2} ( t )} d t + \int \frac{7 sin ( t )}{cos ^{2} ( t )} d t

\int \frac{8}{cos ^{2} ( t )} d t = 8 tan (t)

\int \frac{7 sin ( t )}{cos ^{2} ( t )} d t = 7 sec (t)

= 8 tan (t) + 7 sec (t)

Add a constant to the solution

= 8 tan (t) + 7 sec (t) + C

f (x) = 2 x^{2} + x - 1, at a = - 1

y^{^{'}} = 5 y^{2} + x y^{2}, y (0) = 1

\frac{d}{d x} (\frac{- 8 x - 2}{9 x - 4})

\int (cos (2 x))^{4} d x

\frac{d y}{d x}, y^{3} = x^{5}

What is the integral of sec(t)(8sec(t)+7tan(t)) ?
The integral of sec(t)(8sec(t)+7tan(t)) is 8tan(t)+7sec(t)+C