integral de (sin^3(x))/(1+cos^2(x))

Solución

\int \frac{sin ^{3} ( x )}{1 + cos ^{2} ( x )} d x

- 2 arctan (cos (x)) + cos (x) + C

Pasos de solución

\int \frac{sin ^{3} ( x )}{1 + cos ^{2} ( x )} d x

Re-escribir usando identidades trigonométricas

= \int \frac{sin ^{3} ( x )}{- sin ^{2} ( x ) + 2} d x

Aplicar integración por sustitución trigonométrica

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

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

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

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

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

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

\int 1 d u = u

= - (2 arctan (u) - u)

Sustituir en la ecuación

u = cos (x)

= - (2 arctan (cos (x)) - cos (x))

Simplificar

= - 2 arctan (cos (x)) + cos (x)

Agregar una constante a la solución

= - 2 arctan (cos (x)) + cos (x) + C

\int (2 x^{2} + 7)^{2} d x

\int 2 x 5 x + 1 d x

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

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\int ln (sin (2 x)) d x