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

The integral of sec(θ)(sec(θ)+tan(θ)) is tan(θ)+sec(θ)+C

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integral of sec(θ)(sec(θ)+tan(θ))

\int sec (θ) (sec (θ) + tan (θ)) d θ

tan (θ) + sec (θ) + C

Solution steps

\int sec (θ) (sec (θ) + tan (θ)) d θ

Rewrite using trig identities

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

Expand

\frac{1 + tan ( θ ) cos ( θ )}{cos ^{2} ( θ )} : \frac{1}{cos ^{2} ( θ )} + \frac{tan ( θ )}{cos ( θ )}

= \int \frac{1}{cos ^{2} ( θ )} + \frac{tan ( θ )}{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

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

\int \frac{1}{cos ^{2} ( θ )} d θ = tan (θ)

\int \frac{tan ( θ )}{cos ( θ )} d θ = sec (θ)

= tan (θ) + sec (θ)

Add a constant to the solution

= tan (θ) + sec (θ) + C

x \to 0 lim (\frac{x}{sin ( 12 x )})

\int \frac{60}{2 x + 5} - 4 d x

\int_{1}^{8} x^{- \frac{2}{3}} d x

\int \frac{36 - x ^{2}}{x ^{2}} d x

f (x) = \frac{x ^{2} 3 x}{6}

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