integral de (0.25)/((0.25^2+x^2)^{3/2)}

Solución

\int \frac{0.25}{( 0.2 5 ^{2} + x ^{2} ) ^{\frac{3}{2}}} d x

\frac{16 x}{( 1 + 16 x ^{2} ) ^{\frac{1}{2}}} + C

Pasos de solución

\int \frac{0.25}{( 0.2 5 ^{2} + x ^{2} ) ^{\frac{3}{2}}} d x

Sacar la constante:

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

= 0.25 \cdot \int \frac{1}{( 0.2 5 ^{2} + x ^{2} ) ^{\frac{3}{2}}} d x

Aplicar integración por sustitución trigonométrica

: \int \frac{0.25}{0.062 5 ^{\frac{3}{2}} sec ( u )} d u

= 0.25 \cdot \int \frac{0.25}{0.062 5 ^{\frac{3}{2}} sec ( u )} d u

Sacar la constante:

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

= 0.25 \cdot \frac{0.25}{0.062 5 ^{\frac{3}{2}}} \cdot \int \frac{1}{sec ( u )} d u

Simplificar

= 0.25 \cdot 16 \cdot \int \frac{1}{sec ( u )} d u

Re-escribir usando identidades trigonométricas

= 0.25 \cdot 16 \cdot \int cos (u) d u

Aplicar la regla de integración:

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

= 0.25 \cdot 16 sin (u)

Sustituir en la ecuación

u = arctan (\frac{1}{0.25} x)

= 0.25 \cdot 16 sin (arctan (\frac{1}{0.25} x))

Simplificar

0.25 \cdot 16 sin (arctan (\frac{1}{0.25} x)) : \frac{16 x}{( 1 + 16 x ^{2} ) ^{\frac{1}{2}}}

= \frac{16 x}{( 1 + 16 x ^{2} ) ^{\frac{1}{2}}}

Agregar una constante a la solución

= \frac{16 x}{( 1 + 16 x ^{2} ) ^{\frac{1}{2}}} + C

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