What is the integral of sec(x)(9sec(x)+7tan(x)) ?

The integral of sec(x)(9sec(x)+7tan(x)) is 9tan(x)+7sec(x)+C

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integral of sec(x)(9sec(x)+7tan(x))

\int sec (x) (9 sec (x) + 7 tan (x)) d x

9 tan (x) + 7 sec (x) + C

Solution steps

\int sec (x) (9 sec (x) + 7 tan (x)) d x

Rewrite using trig identities

= \int \frac{9 + 7 sin ( x )}{cos ^{2} ( x )} d x

Expand

\frac{9 + 7 sin ( x )}{cos ^{2} ( x )} : \frac{9}{cos ^{2} ( x )} + \frac{7 sin ( x )}{cos ^{2} ( x )}

= \int \frac{9}{cos ^{2} ( x )} + \frac{7 sin ( x )}{cos ^{2} ( x )} d x

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{9}{cos ^{2} ( x )} d x + \int \frac{7 sin ( x )}{cos ^{2} ( x )} d x

\int \frac{9}{cos ^{2} ( x )} d x = 9 tan (x)

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

= 9 tan (x) + 7 sec (x)

Add a constant to the solution

= 9 tan (x) + 7 sec (x) + C

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\int - 0.1 x d x

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

What is the integral of sec(x)(9sec(x)+7tan(x)) ?
The integral of sec(x)(9sec(x)+7tan(x)) is 9tan(x)+7sec(x)+C