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integral (sin^4(x))/(cos^3(x))

Solución

integral

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

sin (x) - \frac{3}{4} ln ∣ sin (x) + 1 ∣ - \frac{1}{4 ( sin ( x ) + 1 )} + \frac{3}{4} ln ∣ sin (x) - 1 ∣ - \frac{1}{4 ( sin ( x ) - 1 )} + C

Pasos de solución

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

Re-escribir usando identidades trigonométricas

= \int tan^{3} (x) sin (x) d x

Aplicar integración por sustitución

: \int \frac{u ^{4}}{( 1 - u ^{2} ) ^{\frac{3}{2}} 1 - u ^{2}} d u

= \int \frac{u ^{4}}{( 1 - u ^{2} ) ^{\frac{3}{2}} 1 - u ^{2}} d u

(1 - u^{2})^{\frac{3}{2}} = (1 - u^{2}) 1 - u^{2}

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

(1 - u^{2}) 1 - u^{2} 1 - u^{2} = (1 - u^{2})^{2}

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

Tomar la fracción parcial de

\frac{u ^{4}}{( 1 - u ^{2} ) ^{2}} : 1 - \frac{3}{4 ( u + 1 )} + \frac{1}{4 ( u + 1 ) ^{2}} + \frac{3}{4 ( u - 1 )} + \frac{1}{4 ( u - 1 ) ^{2}}

= \int 1 - \frac{3}{4 ( u + 1 )} + \frac{1}{4 ( u + 1 ) ^{2}} + \frac{3}{4 ( u - 1 )} + \frac{1}{4 ( u - 1 ) ^{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 1 d u - \int \frac{3}{4 ( u + 1 )} d u + \int \frac{1}{4 ( u + 1 ) ^{2}} d u + \int \frac{3}{4 ( u - 1 )} d u + \int \frac{1}{4 ( u - 1 ) ^{2}} d u

\int 1 d u = u

\int \frac{3}{4 ( u + 1 )} d u = \frac{3}{4} ln ∣ u + 1 ∣

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

\int \frac{3}{4 ( u - 1 )} d u = \frac{3}{4} ln ∣ u - 1 ∣

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

= u - \frac{3}{4} ln ∣ u + 1 ∣ - \frac{1}{4 ( u + 1 )} + \frac{3}{4} ln ∣ u - 1 ∣ - \frac{1}{4 ( u - 1 )}

Sustituir en la ecuación

u = sin (x)

= sin (x) - \frac{3}{4} ln ∣ sin (x) + 1 ∣ - \frac{1}{4 ( sin ( x ) + 1 )} + \frac{3}{4} ln ∣ sin (x) - 1 ∣ - \frac{1}{4 ( sin ( x ) - 1 )}

Agregar una constante a la solución

= sin (x) - \frac{3}{4} ln ∣ sin (x) + 1 ∣ - \frac{1}{4 ( sin ( x ) + 1 )} + \frac{3}{4} ln ∣ sin (x) - 1 ∣ - \frac{1}{4 ( sin ( x ) - 1 )} + C

- 7 x

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

\frac{( 3 x )}{( 1 + x - 2 x ^{2} )}

\frac{x}{( x ^{2} - x + 10 )}

\frac{1}{ln ( 1 + 3 x )}