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integral de (sqrt(4x^2+1))/x

Solución

\int \frac{4 x ^{2} + 1}{x} d x

Solución

- \frac{1}{2} ln 4 x^{2} + 1 + 1 + \frac{1}{2} ln 4 x^{2} + 1 - 1 + 4 x^{2} + 1 + C

Pasos de solución

\int \frac{4 x ^{2} + 1}{x} d x

Aplicar integración por sustitución

: \int \frac{4 u + 1}{2 u} d u

= \int \frac{4 u + 1}{2 u} d u

Sacar la constante:

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

= \frac{1}{2} \cdot \int \frac{4 u + 1}{u} d u

Aplicar integración por sustitución

: \int \frac{2 v ^{2}}{v ^{2} - 1} d v

= \frac{1}{2} \cdot \int \frac{2 v ^{2}}{v ^{2} - 1} d v

Sacar la constante:

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

= \frac{1}{2} \cdot 2 \cdot \int \frac{v ^{2}}{v ^{2} - 1} d v

Factorizar

v^{2} - 1 : - (- v^{2} + 1)

= \frac{1}{2} \cdot 2 \cdot \int \frac{v ^{2}}{- ( - v ^{2} + 1 )} d v

Sacar la constante:

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

= \frac{1}{2} \cdot 2 (- \int \frac{v ^{2}}{- v ^{2} + 1} d v)

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= \frac{1}{2} \cdot 2 (- \int \frac{1}{- v ^{2} + 1} - 1 d v)

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 2 (- (\int \frac{1}{- v ^{2} + 1} d v - \int 1 d v))

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

\int 1 d v = v

= \frac{1}{2} \cdot 2 (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v))

Sustitución hacia atrás

= \frac{1}{2} \cdot 2 (- (\frac{ln 4 x ^{2} + 1 + 1}{2} - \frac{ln 4 x ^{2} + 1 - 1}{2} - 4 x^{2} + 1))

Simplificar

\frac{1}{2} \cdot 2 (- (\frac{ln 4 x ^{2} + 1 + 1}{2} - \frac{ln 4 x ^{2} + 1 - 1}{2} - 4 x^{2} + 1)) : - \frac{1}{2} ln 4 x^{2} + 1 + 1 + \frac{1}{2} ln 4 x^{2} + 1 - 1 + 4 x^{2} + 1

= - \frac{1}{2} ln 4 x^{2} + 1 + 1 + \frac{1}{2} ln 4 x^{2} + 1 - 1 + 4 x^{2} + 1

Agregar una constante a la solución

= - \frac{1}{2} ln 4 x^{2} + 1 + 1 + \frac{1}{2} ln 4 x^{2} + 1 - 1 + 4 x^{2} + 1 + C

Ejemplos populares

integral de 1/(x^2+b^2)

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

integral de (3x^2+6x+1)/x

\int \frac{3 x ^{2} + 6 x + 1}{x} d x

integral de sinh(7x)

\int sinh (7 x) d x

integral de ((x^2-2)(x^2+6))/(3x^3)

\int \frac{( x ^{2} - 2 ) ( x ^{2} + 6 )}{3 x ^{3}} d x

integral de e^xsin(2x+1)

\int e^{x} sin (2 x + 1) d x