What is the integral of sec(t)(6sec(t)+7tan(t)) ?

The integral of sec(t)(6sec(t)+7tan(t)) is 6tan(t)+7sec(t)+C

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integral of sec(t)(6sec(t)+7tan(t))

\int sec (t) (6 sec (t) + 7 tan (t)) d t

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

Solution steps

\int sec (t) (6 sec (t) + 7 tan (t)) d t

Rewrite using trig identities

= \int \frac{6 + 7 sin ( t )}{cos ^{2} ( t )} d t

Expand

\frac{6 + 7 sin ( t )}{cos ^{2} ( t )} : \frac{6}{cos ^{2} ( t )} + \frac{7 sin ( t )}{cos ^{2} ( t )}

= \int \frac{6}{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{6}{cos ^{2} ( t )} d t + \int \frac{7 sin ( t )}{cos ^{2} ( t )} d t

\int \frac{6}{cos ^{2} ( t )} d t = 6 tan (t)

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

= 6 tan (t) + 7 sec (t)

Add a constant to the solution

= 6 tan (t) + 7 sec (t) + C

\frac{d}{d x} (\frac{7}{x ^{2} + 1})

\int x^{3} x^{2} + 32 d x

\frac{d}{d x} (3 cos (x) + 2 sin (x))

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

\int e^{k x} d x

What is the integral of sec(t)(6sec(t)+7tan(t)) ?
The integral of sec(t)(6sec(t)+7tan(t)) is 6tan(t)+7sec(t)+C