English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral de (3^{x-1})/(sec(3^x))

Solución

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

\frac{1}{3 ln ( 3 )} sin (3^{x}) + C

Pasos de solución

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

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

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

Sacar la constante:

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

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

Aplicar las leyes de los exponentes:

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

= \frac{1}{3} \cdot \int \frac{3 ^{x}}{sec ( 3 ^{x} )} d x

Utilizar la identidad trigonométrica básica:

\frac{1}{sec ( x )} = cos (x)

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

Aplicar integración por sustitución

: \int \frac{cos ( u )}{ln ( 3 )} d u

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

Sacar la constante:

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

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

Aplicar la regla de integración:

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

= \frac{1}{3} \cdot \frac{1}{ln ( 3 )} sin (u)

Sustituir en la ecuación

u = 3^{x}

= \frac{1}{3} \cdot \frac{1}{ln ( 3 )} sin (3^{x})

Simplificar

\frac{1}{3} \cdot \frac{1}{ln ( 3 )} sin (3^{x}) : \frac{1}{3 ln ( 3 )} sin (3^{x})

= \frac{1}{3 ln ( 3 )} sin (3^{x})

Agregar una constante a la solución

= \frac{1}{3 ln ( 3 )} sin (3^{x}) + C

\int sin (6 x) sin (8 x) d x

\int \frac{2}{x ^{2} - 2 x + 1} d x

\int \frac{x}{( x ^{2} + 5 ) ^{\frac{7}{2}}} d x

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

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