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integral de (sec(θ)tan(θ))/(3+2sec(θ))

Solución

\int \frac{sec ( θ ) tan ( θ )}{3 + 2 sec ( θ )} d θ

\frac{1}{2} ln ∣ 3 cos (θ) + 2 ∣ - \frac{1}{2} ln (3 ∣ cos (θ) ∣) + C

Pasos de solución

\int \frac{sec ( θ ) tan ( θ )}{3 + 2 sec ( θ )} d θ

Re-escribir usando identidades trigonométricas

= \int \frac{sin ( θ )}{cos ( θ ) ( 3 cos ( θ ) + 2 )} d θ

Simplificar

\frac{sin ( θ )}{cos ( θ ) ( 3 cos ( θ ) + 2 )} : \frac{tan ( θ )}{3 cos ( θ ) + 2}

= \int \frac{tan ( θ )}{3 cos ( θ ) + 2} d θ

Aplicar integración por sustitución

: \int - \frac{1}{u ( 3 u + 2 )} d u

= \int - \frac{1}{u ( 3 u + 2 )} d u

Sacar la constante:

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

= - \int \frac{1}{u ( 3 u + 2 )} d u

Aplicar integración por sustitución

: \int \frac{1}{v ( v - 2 )} d v

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

Completar el cuadrado

v (v - 2) : (v - 1)^{2} - 1

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

Aplicar integración por sustitución

: \int \frac{1}{w ^{2} - 1} d w

= - \int \frac{1}{w ^{2} - 1} d w

Factorizar

w^{2} - 1 : - (- w^{2} + 1)

= - \int \frac{1}{- ( - w ^{2} + 1 )} d w

Sacar la constante:

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

= - (- \int \frac{1}{- w ^{2} + 1} d w)

Aplicar la regla de integración:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= - (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

Sustitución hacia atrás

= - (- (\frac{ln ∣ 3 cos ( θ ) + 2 - 1 + 1 ∣}{2} - \frac{ln ∣ 3 cos ( θ ) + 2 - 1 - 1 ∣}{2}))

Simplificar

- (- (\frac{ln ∣ 3 cos ( θ ) + 2 - 1 + 1 ∣}{2} - \frac{ln ∣ 3 cos ( θ ) + 2 - 1 - 1 ∣}{2})) : \frac{1}{2} ln ∣ 3 cos (θ) + 2 ∣ - \frac{1}{2} ln (3 ∣ cos (θ) ∣)

= \frac{1}{2} ln ∣ 3 cos (θ) + 2 ∣ - \frac{1}{2} ln (3 ∣ cos (θ) ∣)

Agregar una constante a la solución

= \frac{1}{2} ln ∣ 3 cos (θ) + 2 ∣ - \frac{1}{2} ln (3 ∣ cos (θ) ∣) + C

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