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Popular >

integral de (3x)/(25x^4-4)

Solución

\int \frac{3 x}{25 x ^{4} - 4} d x

Solución

- \frac{3}{40} (ln \frac{5}{2} x^{2} + 1 - ln \frac{5}{2} x^{2} - 1) + C

Pasos de solución

\int \frac{3 x}{25 x ^{4} - 4} d x

Sacar la constante:

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

= 3 \cdot \int \frac{x}{25 x ^{4} - 4} d x

Factorizar

25 x^{4} - 4 : - (- 25 x^{4} + 4)

= 3 \cdot \int \frac{x}{- ( - 25 x ^{4} + 4 )} d x

Sacar la constante:

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

= 3 (- \int \frac{x}{- 25 x ^{4} + 4} d x)

Aplicar integración por sustitución

: \int \frac{1}{2 ( - 25 u ^{2} + 4 )} d u

= 3 (- \int \frac{1}{2 ( - 25 u ^{2} + 4 )} d u)

Sacar la constante:

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

= 3 (- \frac{1}{2} \cdot \int \frac{1}{- 25 u ^{2} + 4} d u)

Aplicar integración por sustitución

= 3 (- \frac{1}{2} \cdot \int \frac{1}{10 ( - v ^{2} + 1 )} d v)

Sacar la constante:

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

= 3 (- \frac{1}{2} \cdot \frac{1}{10} \cdot \int \frac{1}{- v ^{2} + 1} d v)

Aplicar la regla de integración:

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

= 3 (- \frac{1}{2} \cdot \frac{1}{10} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Sustitución hacia atrás

= 3 (- \frac{1}{2} \cdot \frac{1}{10} (\frac{ln \frac{5}{2} x ^{2} + 1}{2} - \frac{ln \frac{5}{2} x ^{2} - 1}{2}))

Simplificar

3 (- \frac{1}{2} \cdot \frac{1}{10} (\frac{ln \frac{5}{2} x ^{2} + 1}{2} - \frac{ln \frac{5}{2} x ^{2} - 1}{2})) : - \frac{3}{40} (ln \frac{5}{2} x^{2} + 1 - ln \frac{5}{2} x^{2} - 1)

= - \frac{3}{40} (ln \frac{5}{2} x^{2} + 1 - ln \frac{5}{2} x^{2} - 1)

Agregar una constante a la solución

= - \frac{3}{40} (ln \frac{5}{2} x^{2} + 1 - ln \frac{5}{2} x^{2} - 1) + C

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\frac{d}{d x} (\frac{20 x + x ^{2}}{( 10 + x ) ^{2}})

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\int 2 r^{3} 1 - r^{2} d r