integral de (tan^2(2x))/(sec(2x))

Solución

\int \frac{tan ^{2} ( 2 x )}{sec ( 2 x )} d x

\frac{1}{2} (ln ∣ tan (2 x) + sec (2 x) ∣ - sin (2 x)) + C

Pasos de solución

\int \frac{tan ^{2} ( 2 x )}{sec ( 2 x )} d x

Re-escribir usando identidades trigonométricas

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

Aplicar integración por sustitución

: \int \frac{sin ( u ) tan ( u )}{2} d u

= \int \frac{sin ( u ) tan ( u )}{2} d u

Sacar la constante:

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

= \frac{1}{2} \cdot \int sin (u) tan (u) d u

Re-escribir usando identidades trigonométricas

= \frac{1}{2} \cdot \int \frac{1 - cos ^{2} ( u )}{cos ( u )} d u

Expandir

\frac{1 - cos ^{2} ( u )}{cos ( u )} : \frac{1}{cos ( u )} - cos (u)

= \frac{1}{2} \cdot \int \frac{1}{cos ( u )} - cos (u) d u

Aplicar la regla de la suma:

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

= \frac{1}{2} (\int \frac{1}{cos ( u )} d u - \int cos (u) d u)

\int \frac{1}{cos ( u )} d u = ln ∣ tan (u) + sec (u) ∣

\int cos (u) d u = sin (u)

= \frac{1}{2} (ln ∣ tan (u) + sec (u) ∣ - sin (u))

Sustituir en la ecuación

u = 2 x

= \frac{1}{2} (ln ∣ tan (2 x) + sec (2 x) ∣ - sin (2 x))

Agregar una constante a la solución

= \frac{1}{2} (ln ∣ tan (2 x) + sec (2 x) ∣ - sin (2 x)) + C

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