integral de (sin(x))/(sin^2(x)+1)

Solución

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

- \frac{1}{2 2} (ln \frac{cos ( x )}{2} + 1 - ln \frac{cos ( x )}{2} - 1) + C

Pasos de solución

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

Re-escribir usando identidades trigonométricas

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

Aplicar integración por sustitución

: \int - \frac{1}{- u ^{2} + 2} d u

= \int - \frac{1}{- u ^{2} + 2} d u

Sacar la constante:

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

= - \int \frac{1}{- u ^{2} + 2} d u

Aplicar integración por sustitución

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

Sacar la constante:

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

= - \frac{1}{2} \cdot \int \frac{1}{- v ^{2} + 1} d v

Aplicar la regla de integración:

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

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

Sustitución hacia atrás

= - \frac{1}{2} \frac{ln \frac{c o s ( x )}{2} + 1}{2} - \frac{ln \frac{c o s ( x )}{2} - 1}{2}

Simplificar

- \frac{1}{2} \frac{ln \frac{c o s ( x )}{2} + 1}{2} - \frac{ln \frac{c o s ( x )}{2} - 1}{2} : - \frac{1}{2 2} (ln \frac{cos ( x )}{2} + 1 - ln \frac{cos ( x )}{2} - 1)

= - \frac{1}{2 2} (ln \frac{cos ( x )}{2} + 1 - ln \frac{cos ( x )}{2} - 1)

Agregar una constante a la solución

= - \frac{1}{2 2} (ln \frac{cos ( x )}{2} + 1 - ln \frac{cos ( x )}{2} - 1) + C

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