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integral de 4/(t-sqrt(t+7)+1)

Solución

\int \frac{4}{t - t + 7 + 1} d t

16 (\frac{1}{4} ln 4 t - 4 t + 7 + 4 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} t + 7 - ln - \frac{6}{5} + \frac{2}{5} t + 7)) + C

Pasos de solución

\int \frac{4}{t - t + 7 + 1} d t

Sacar la constante:

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

= 4 \cdot \int \frac{1}{t - t + 7 + 1} d t

Aplicar integración por sustitución

: \int \frac{2 u}{u ^{2} - u - 6} d u

= 4 \cdot \int \frac{2 u}{u ^{2} - u - 6} d u

Sacar la constante:

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

= 4 \cdot 2 \cdot \int \frac{u}{u ^{2} - u - 6} d u

Completar el cuadrado

u^{2} - u - 6 : (u - \frac{1}{2})^{2} - \frac{25}{4}

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

Aplicar integración por sustitución

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

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

Sacar la constante:

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

= 4 \cdot 2 \cdot 2 \cdot \int \frac{2 v + 1}{4 v ^{2} - 25} d v

Expandir

\frac{2 v + 1}{4 v ^{2} - 25} : \frac{2 v}{4 v ^{2} - 25} + \frac{1}{4 v ^{2} - 25}

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

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

\int \frac{2 v}{4 v ^{2} - 25} d v = \frac{1}{4} ln 4 v^{2} - 25

\int \frac{1}{4 v ^{2} - 25} d v = - \frac{1}{20} (ln \frac{2}{5} v + 1 - ln \frac{2}{5} v - 1)

= 4 \cdot 2 \cdot 2 (\frac{1}{4} ln 4 v^{2} - 25 - \frac{1}{20} (ln \frac{2}{5} v + 1 - ln \frac{2}{5} v - 1))

Sustitución hacia atrás

= 4 \cdot 2 \cdot 2 (\frac{1}{4} ln 4 (t + 7 - \frac{1}{2})^{2} - 25 - \frac{1}{20} (ln \frac{2}{5} (t + 7 - \frac{1}{2}) + 1 - ln \frac{2}{5} (t + 7 - \frac{1}{2}) - 1))

Simplificar

4 \cdot 2 \cdot 2 (\frac{1}{4} ln 4 (t + 7 - \frac{1}{2})^{2} - 25 - \frac{1}{20} (ln \frac{2}{5} (t + 7 - \frac{1}{2}) + 1 - ln \frac{2}{5} (t + 7 - \frac{1}{2}) - 1)) : 16 (\frac{1}{4} ln 4 t - 4 t + 7 + 4 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} t + 7 - ln - \frac{6}{5} + \frac{2}{5} t + 7))

= 16 (\frac{1}{4} ln 4 t - 4 t + 7 + 4 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} t + 7 - ln - \frac{6}{5} + \frac{2}{5} t + 7))

Agregar una constante a la solución

= 16 (\frac{1}{4} ln 4 t - 4 t + 7 + 4 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} t + 7 - ln - \frac{6}{5} + \frac{2}{5} t + 7)) + C

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