integral de (sec^4(x))/(tan^3(x))

Solución

\int \frac{sec ^{4} ( x )}{tan ^{3} ( x )} d x

- \frac{1}{2} cot^{2} (x) + ln ∣ tan (x) ∣ + C

Pasos de solución

\int \frac{sec ^{4} ( x )}{tan ^{3} ( x )} d x

Aplicar las leyes de los exponentes:

\frac{1}{a ^{b}} = a^{- b}

\frac{sec ^{4} ( x )}{tan ^{3} ( x )} = (sec^{4} (x)) tan^{- 3} (x)

= \int sec^{4} (x) tan^{- 3} (x) d x

Re-escribir usando identidades trigonométricas

= \int (1 + tan^{2} (x)) sec^{2} (x) tan^{- 3} (x) d x

Aplicar integración por sustitución

: \int \frac{1 + u ^{2}}{u ^{3}} d u

= \int \frac{1 + u ^{2}}{u ^{3}} d u

Expandir

\frac{1 + u ^{2}}{u ^{3}} : \frac{1}{u ^{3}} + \frac{1}{u}

= \int \frac{1}{u ^{3}} + \frac{1}{u} 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 ^{3}} d u + \int \frac{1}{u} d u

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

\int \frac{1}{u} d u = ln ∣ u ∣

= - \frac{1}{2 u ^{2}} + ln ∣ u ∣

Sustituir en la ecuación

u = tan (x)

= - \frac{1}{2 tan ^{2} ( x )} + ln ∣ tan (x) ∣

Simplificar

- \frac{1}{2 tan ^{2} ( x )} + ln ∣ tan (x) ∣ : - \frac{1}{2} cot^{2} (x) + ln ∣ tan (x) ∣

= - \frac{1}{2} cot^{2} (x) + ln ∣ tan (x) ∣

Agregar una constante a la solución

= - \frac{1}{2} cot^{2} (x) + ln ∣ tan (x) ∣ + C

\int (x - 5) cos (x) d x

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sin (x^{4})