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integral de (x^2cot^2(x^3))/(2csc(x^3))

Solución

\int \frac{x ^{2} cot ^{2} ( x ^{3} )}{2 csc ( x ^{3} )} d x

\frac{1}{6} (ln tan (\frac{x ^{3}}{2}) + cos (x^{3})) + C

Pasos de solución

\int \frac{x ^{2} cot ^{2} ( x ^{3} )}{2 csc ( x ^{3} )} d x

Sacar la constante:

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

= \frac{1}{2} \cdot \int \frac{x ^{2} cot ^{2} ( x ^{3} )}{csc ( x ^{3} )} d x

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica básica:

\frac{cos ( x )}{sin ( x )} = cot (x)

= \frac{1}{2} \cdot \int x^{2} cos (x^{3}) cot (x^{3}) d x

Aplicar integración por sustitución

: \int \frac{u cos ( u ^{\frac{3}{2}} ) cot ( u ^{\frac{3}{2}} )}{2 x} d u

= \frac{1}{2} \cdot \int \frac{cos ( u ) cot ( u )}{3} d u

Sacar la constante:

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

= \frac{1}{2} \cdot \frac{1}{3} \cdot \int cos (u) cot (u) d u

Re-escribir usando identidades trigonométricas

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

Expandir

\frac{1 - sin ^{2} ( u )}{sin ( u )} : \frac{1}{sin ( u )} - sin (u)

= \frac{1}{2} \cdot \frac{1}{3} \cdot \int \frac{1}{sin ( u )} - sin (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

= \frac{1}{2} \cdot \frac{1}{3} (\int \frac{1}{sin ( u )} d u - \int sin (u) d u)

\int \frac{1}{sin ( u )} d u = ln tan (\frac{u}{2})

\int sin (u) d u = - cos (u)

= \frac{1}{2} \cdot \frac{1}{3} (ln tan (\frac{u}{2}) - (- cos (u)))

Sustituir en la ecuación

u = x^{3}

= \frac{1}{2} \cdot \frac{1}{3} (ln tan (\frac{x ^{3}}{2}) - (- cos (x^{3})))

Simplificar

\frac{1}{2} \cdot \frac{1}{3} (ln tan (\frac{x ^{3}}{2}) - (- cos (x^{3}))) : \frac{1}{6} (ln tan (\frac{x ^{3}}{2}) + cos (x^{3}))

= \frac{1}{6} (ln tan (\frac{x ^{3}}{2}) + cos (x^{3}))

Agregar una constante a la solución

= \frac{1}{6} (ln tan (\frac{x ^{3}}{2}) + cos (x^{3})) + C

\int e^{l n (x)} \cdot e^{2 x} (x^{- 1} + 2) d x

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\int e^{- 2 x} cos (5 x) d x

\int \frac{ln ( 5 x )}{x ^{2}} d x

\int \frac{3}{x ( x ^{2} + 5 ) ^{2}} d x