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

Solución

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

\frac{1}{2 2} arctan (2 cot (x)) - \frac{1}{2} cot (x) + C

Pasos de solución

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

Aplicar integración por sustitución trigonométrica

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

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

Sacar la constante:

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

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

Aplicar integración por sustitución

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

\frac{v ^{2}}{v ^{2} + 1} = - \frac{1}{v ^{2} + 1} + 1

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

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

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

\int 1 d v = v

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

Sustitución hacia atrás

= - \frac{1}{2 2} (- arctan (2 cot (x)) + 2 cot (x))

Simplificar

- \frac{1}{2 2} (- arctan (2 cot (x)) + 2 cot (x)) : \frac{1}{2 2} arctan (2 cot (x)) - \frac{1}{2} cot (x)

= \frac{1}{2 2} arctan (2 cot (x)) - \frac{1}{2} cot (x)

Agregar una constante a la solución

= \frac{1}{2 2} arctan (2 cot (x)) - \frac{1}{2} cot (x) + C

y = - x, y = 2 - x, y = 0

y = \frac{3 x ^{2}}{2 x + 4}

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

\frac{\partial}{\partial x} (2 A y (L - x))

\int sec^{2} (7 - 4 x) d x