integral de xarctan(sqrt(7x))

Solución

\int x arctan (7 x) d x

\frac{1}{2} x^{2} arctan (7 x) - \frac{1}{294} (77 x^{\frac{3}{2}} - 37 x + 3 arctan (7 x)) + C

Pasos de solución

\int x arctan (7 x) d x

Aplicar integración por sustitución

: \int \frac{u arctan ( u )}{49} d u

= \int \frac{u arctan ( u )}{49} d u

Sacar la constante:

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

= \frac{1}{49} \cdot \int u arctan (u) d u

Aplicar integración por sustitución

: \int 2 v^{3} arctan (v) d v

= \frac{1}{49} \cdot \int 2 v^{3} arctan (v) d v

Sacar la constante:

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

= \frac{1}{49} \cdot 2 \cdot \int v^{3} arctan (v) d v

Aplicar integración por partes

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

= \frac{1}{49} \cdot 2 (\frac{1}{4} v^{4} arctan (v) - \int \frac{v ^{4}}{4 ( v ^{2} + 1 )} d v)

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

= \frac{1}{49} \cdot 2 (\frac{1}{4} v^{4} arctan (v) - \frac{1}{4} (\frac{v ^{3}}{3} - v + arctan (v)))

Sustitución hacia atrás

= \frac{1}{49} \cdot 2 (\frac{1}{4} (7 x)^{4} arctan (7 x) - \frac{1}{4} (\frac{( 7 x ) ^{3}}{3} - 7 x + arctan (7 x)))

Simplificar

\frac{1}{49} \cdot 2 (\frac{1}{4} (7 x)^{4} arctan (7 x) - \frac{1}{4} (\frac{( 7 x ) ^{3}}{3} - 7 x + arctan (7 x))) : \frac{1}{2} x^{2} arctan (7 x) - \frac{1}{294} (77 x^{\frac{3}{2}} - 37 x + 3 arctan (7 x))

= \frac{1}{2} x^{2} arctan (7 x) - \frac{1}{294} (77 x^{\frac{3}{2}} - 37 x + 3 arctan (7 x))

Agregar una constante a la solución

= \frac{1}{2} x^{2} arctan (7 x) - \frac{1}{294} (77 x^{\frac{3}{2}} - 37 x + 3 arctan (7 x)) + C

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