integral de 4 a 5 de x/(x^2+4x+20)

Solución

\int_{4}^{5} \frac{x}{x ^{2} + 4 x + 20} d x

\frac{1}{2} (ln (\frac{5}{4}) - arctan (\frac{7}{4}) + arctan (\frac{3}{2}))

Decimal

0.07714 \dots

Pasos de solución

\int_{4}^{5} \frac{x}{x ^{2} + 4 x + 20} d x

Completar el cuadrado

x^{2} + 4 x + 20 : (x + 2)^{2} + 16

= \int_{4}^{5} \frac{x}{( x + 2 ) ^{2} + 16} d x

Aplicar integración por sustitución

= \int_{6}^{7} \frac{u - 2}{u ^{2} + 16} d u

Aplicar integración por sustitución

= \int_{\frac{3}{2}}^{\frac{7}{4}} \frac{2 v - 1}{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 \int_{\frac{3}{2}}^{\frac{7}{4}} \frac{2 v - 1}{v ^{2} + 1} d v

Expandir

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

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} (\int_{\frac{3}{2}}^{\frac{7}{4}} \frac{2 v}{v ^{2} + 1} d v - \int_{\frac{3}{2}}^{\frac{7}{4}} \frac{1}{v ^{2} + 1} d v)

\int_{\frac{3}{2}}^{\frac{7}{4}} \frac{2 v}{v ^{2} + 1} d v = ln (\frac{5}{4})

\int_{\frac{3}{2}}^{\frac{7}{4}} \frac{1}{v ^{2} + 1} d v = arctan (\frac{7}{4}) - arctan (\frac{3}{2})

= \frac{1}{2} (ln (\frac{5}{4}) - (arctan (\frac{7}{4}) - arctan (\frac{3}{2})))

Simplificar

= \frac{1}{2} (ln (\frac{5}{4}) - arctan (\frac{7}{4}) + arctan (\frac{3}{2}))