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

The integral of sec(θ)(sec(θ)+tan(x)) is sec(x)(cos(x)tan(θ)+sin(x)ln|tan(θ)+sec(θ)|)+C

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integral of sec(θ)(sec(θ)+tan(x))

\int sec (θ) (sec (θ) + tan (x)) d θ

sec (x) (cos (x) tan (θ) + sin (x) ln ∣ tan (θ) + sec (θ) ∣) + C

Solution steps

\int sec (θ) (sec (θ) + tan (x)) d θ

Rewrite using trig identities

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int \frac{sin ( x )}{cos ( θ )} d θ = sin (x) ln ∣ tan (θ) + sec (θ) ∣

= \frac{1}{cos ( x )} (cos (x) tan (θ) + sin (x) ln ∣ tan (θ) + sec (θ) ∣)

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= sec (x) (cos (x) tan (θ) + sin (x) ln ∣ tan (θ) + sec (θ) ∣)

Add a constant to the solution

= sec (x) (cos (x) tan (θ) + sin (x) ln ∣ tan (θ) + sec (θ) ∣) + C

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What is the integral of sec(θ)(sec(θ)+tan(x)) ?
The integral of sec(θ)(sec(θ)+tan(x)) is sec(x)(cos(x)tan(θ)+sin(x)ln|tan(θ)+sec(θ)|)+C