integral de (sin(2x))/(2sin(x)cos^2(x))

Solución

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

ln ∣ tan (x) + sec (x) ∣ + C

Pasos de solución

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

Sacar la constante:

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

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

Re-escribir usando identidades trigonométricas

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

Sacar la constante:

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

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

Re-escribir usando identidades trigonométricas

= \frac{1}{2} \cdot 2 \cdot \int sec (x) d x

Aplicar la regla de integración:

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

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

Simplificar

\frac{1}{2} \cdot 2 ln ∣ tan (x) + sec (x) ∣ : ln ∣ tan (x) + sec (x) ∣

= ln ∣ tan (x) + sec (x) ∣

Agregar una constante a la solución

= ln ∣ tan (x) + sec (x) ∣ + C

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