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Beliebt Rechnen >

integral of sqrt(y^2+x^2)

Lösung

\int y^{2} + x^{2} d x

Lösung

\frac{1}{2} x x^{2} + y^{2} + \frac{y ^{2}}{2} ln \frac{x + y ^{2} + x ^{2}}{∣ y ∣} + C

Schritte zur Lösung

\int y^{2} + x^{2} d x

Trigonometrische Substitution anwenden

= \int y^{2} sec^{3} (u) d u

Entferne die Konstante:

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

= y^{2} \cdot \int sec^{3} (u) d u

Wende integrale Reduktion an

= y^{2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= y^{2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

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

ein

= y^{2} \frac{sec ^{2} ( arctan ( \frac{1}{y} x ) ) sin ( arctan ( \frac{1}{y} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{y} x)) + sec (arctan (\frac{1}{y} x))

Vereinfache

y^{2} \frac{sec ^{2} ( arctan ( \frac{1}{y} x ) ) sin ( arctan ( \frac{1}{y} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{y} x)) + sec (arctan (\frac{1}{y} x)) : \frac{1}{2} x x^{2} + y^{2} + \frac{y ^{2}}{2} ln \frac{x + y ^{2} + x ^{2}}{∣ y ∣}

= \frac{1}{2} x x^{2} + y^{2} + \frac{y ^{2}}{2} ln \frac{x + y ^{2} + x ^{2}}{∣ y ∣}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x x^{2} + y^{2} + \frac{y ^{2}}{2} ln \frac{x + y ^{2} + x ^{2}}{∣ y ∣} + C

Graph

Beliebte Beispiele

integral of 1/((81x^2+16)^2)

\int \frac{1}{( 81 x ^{2} + 16 ) ^{2}} d x

integral of+y^2e^{xy^2}

\int + y^{2} e^{x y^{2}} d x

integral of (sqrt(9x^2-25))/x

\int \frac{9 x ^{2} - 25}{x} d x

integral of (3m+4)^2

\int (3 m + 4)^{2} d m

integral of ((3+x))/((x^2+x+1))

\int \frac{( 3 + x )}{( x ^{2} + x + 1 )} d x