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integral de sin^3(x)cos^{-4}(x)

Solución

\int sin^{3} (x) cos^{- 4} (x) d x

Solución

\frac{1}{3} sec^{3} (x) - sec (x) + C

Pasos de solución

\int sin^{3} (x) cos^{- 4} (x) d x

Simplificar

sin^{3} (x) : sin^{2} (x) sin (x)

= \int sin^{2} (x) sin (x) cos^{- 4} (x) d x

Simplificar

sin^{2} (x) sin (x) cos^{- 4} (x) : \frac{sin ^{3} ( x )}{cos ^{4} ( x )}

= \int \frac{sin ^{3} ( x )}{cos ^{4} ( x )} d x

Re-escribir usando identidades trigonométricas

= \int \frac{( 1 - cos ^{2} ( x ) ) sin ( x )}{cos ^{4} ( x )} d x

Aplicar integración por sustitución

: \int - \frac{1 - u ^{2}}{u ^{4}} d u

= \int - \frac{1 - u ^{2}}{u ^{4}} d u

Expandir

- \frac{1 - u ^{2}}{u ^{4}} : - \frac{1}{u ^{4}} + \frac{1}{u ^{2}}

= \int - \frac{1}{u ^{4}} + \frac{1}{u ^{2}} 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{1}{u ^{4}} d u + \int \frac{1}{u ^{2}} d u

\int \frac{1}{u ^{4}} d u = - \frac{1}{3 u ^{3}}

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

= - (- \frac{1}{3 u ^{3}}) - \frac{1}{u}

Sustituir en la ecuación

u = cos (x)

= - (- \frac{1}{3 cos ^{3} ( x )}) - \frac{1}{cos ( x )}

Simplificar

- (- \frac{1}{3 cos ^{3} ( x )}) - \frac{1}{cos ( x )} : \frac{1}{3} sec^{3} (x) - sec (x)

= \frac{1}{3} sec^{3} (x) - sec (x)

Agregar una constante a la solución

= \frac{1}{3} sec^{3} (x) - sec (x) + C

Ejemplos populares

(20y^2-xy)dx+x^2dy=0

(20 y^{2} - x y) d x + x^{2} d y = 0

derivative (12x^2-24x+24)e^{-8x}derivada de

(12 x^{2} - 24 x + 24) e^{- 8 x}

integral de 1/(t^2sqrt(64-t^2))

\int \frac{1}{t ^{2} 64 - t ^{2}} d t

3 y^{^{''}} + 4 y^{^{'}} = 0

integral de 1 a e de x^4ln(x)

\int_{1}^{e} x^{4} ln (x) d x