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

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

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

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

x + sec (x) + C

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

\int 1 d x = x

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

= x + sec (x)

Add a constant to the solution

= x + sec (x) + C

\frac{1}{1 + s + s ^{2}}

\frac{d}{d x} (5 x sin (x) cos (x))

\int (3 - x) d x

\int_{0}^{l n (π)} 2 x e^{x^{2}} cos (e^{x^{2}}) d x

x \to - 6 - lim (x^{2} + \frac{2}{x + 6})

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