integral de cos(x)sqrt(9+25sin^2(x))

Solución

\int cos (x) 9 + 25 sin^{2} (x) d x

\frac{1}{2} sin (x) 25 sin^{2} (x) + 9 + \frac{9}{10} ln (\frac{1}{3} 5 sin (x) + 9 + 25 sin^{2} (x)) + C

Pasos de solución

\int cos (x) 9 + 25 sin^{2} (x) d x

Aplicar integración por sustitución

: \int 9 + 25 u^{2} d u

= \int 9 + 25 u^{2} d u

Aplicar integración por sustitución trigonométrica

: \int \frac{9}{5} sec^{3} (v) d v

= \int \frac{9}{5} sec^{3} (v) d v

Sacar la constante:

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

= \frac{9}{5} \cdot \int sec^{3} (v) d v

Aplicar la reducción de integrales

= \frac{9}{5} (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= \frac{9}{5} (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Sustitución hacia atrás

= \frac{9}{5} (\frac{sec ^{2} ( arctan ( \frac{5}{3} sin ( x ) ) ) sin ( arctan ( \frac{5}{3} sin ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{5}{3} sin (x))) + sec (arctan (\frac{5}{3} sin (x))))

Simplificar

\frac{9}{5} (\frac{sec ^{2} ( arctan ( \frac{5}{3} sin ( x ) ) ) sin ( arctan ( \frac{5}{3} sin ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{5}{3} sin (x))) + sec (arctan (\frac{5}{3} sin (x)))) : \frac{1}{2} sin (x) 25 sin^{2} (x) + 9 + \frac{9}{10} ln (\frac{1}{3} 5 sin (x) + 9 + 25 sin^{2} (x))

= \frac{1}{2} sin (x) 25 sin^{2} (x) + 9 + \frac{9}{10} ln (\frac{1}{3} 5 sin (x) + 9 + 25 sin^{2} (x))

Agregar una constante a la solución

= \frac{1}{2} sin (x) 25 sin^{2} (x) + 9 + \frac{9}{10} ln (\frac{1}{3} 5 sin (x) + 9 + 25 sin^{2} (x)) + C

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