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Beliebt Rechnen >

integral of 4x^3+2xy^2ln(x^2+y^2)

Lösung

\int 4 x^{3} + 2 x y^{2} ln (x^{2} + y^{2}) d x

Lösung

x^{4} - y^{2} x^{2} + y^{2} x^{2} ln (x^{2} + y^{2}) + y^{4} ln (x^{2} + y^{2}) - 2 y^{4} + C

Schritte zur Lösung

\int 4 x^{3} + 2 x y^{2} ln (x^{2} + y^{2}) d x

Wende U-Substitution an

= \int 2 (u - y^{2}) + y^{2} ln (u) d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 2 (u - y^{2}) d u + \int y^{2} ln (u) d u

\int 2 (u - y^{2}) d u = u^{2} - 2 y^{2} u

\int y^{2} ln (u) d u = y^{2} (u ln (u) - u)

= u^{2} - 2 y^{2} u + y^{2} (u ln (u) - u)

Setze in

u = y^{2} + x^{2}

ein

= (y^{2} + x^{2})^{2} - 2 y^{2} (y^{2} + x^{2}) + y^{2} ((y^{2} + x^{2}) ln (y^{2} + x^{2}) - (y^{2} + x^{2}))

Vereinfache

(y^{2} + x^{2})^{2} - 2 y^{2} (y^{2} + x^{2}) + y^{2} ((y^{2} + x^{2}) ln (y^{2} + x^{2}) - (y^{2} + x^{2})) : x^{4} - y^{2} x^{2} + y^{2} x^{2} ln (x^{2} + y^{2}) + y^{4} ln (x^{2} + y^{2}) - 2 y^{4}

= x^{4} - y^{2} x^{2} + y^{2} x^{2} ln (x^{2} + y^{2}) + y^{4} ln (x^{2} + y^{2}) - 2 y^{4}

Füge eine Konstante zur Lösung hinzu

= x^{4} - y^{2} x^{2} + y^{2} x^{2} ln (x^{2} + y^{2}) + y^{4} ln (x^{2} + y^{2}) - 2 y^{4} + C

Graph

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