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

The integral of sec(x)(tan(x)+cos(x)) is sec(x)+x+C

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integral of sec(x)(tan(x)+cos(x))

\int sec (x) (tan (x) + cos (x)) d x

sec (x) + x + C

Solution steps

\int sec (x) (tan (x) + cos (x)) d x

Rewrite using trig identities

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

Expand

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

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

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

\int 1 d x = x

= sec (x) + x

Add a constant to the solution

= sec (x) + x + C

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