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Beliebt Rechnen >

integral of sqrt(x(x+1/x))

Lösung

\int x (x + \frac{1}{x}) d x

Lösung

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

Schritte zur Lösung

\int x (x + \frac{1}{x}) d x

Multipliziere aus

x (x + \frac{1}{x}) : x^{2} + 1

= \int x^{2} + 1 d x

Trigonometrische Substitution anwenden

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

Wende integrale Reduktion an

= \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) ∣

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

Setze in

u = arctan (x)

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

tangent f(x)=41x-16x^2,\at x=1

t an g e n t f (x) = 41 x - 16 x^{2}, at x = 1

integral of 1/(1+sin(x)cos(x))

\int \frac{1}{1 + sin ( x ) cos ( x )} d x

maclaurin (1+x)^{-6/5}

ma c l a u r in (1 + x)^{- \frac{6}{5}}

tangent y=(1-3x)^{10},(0,1)

t an g e n t y = (1 - 3 x)^{10}, (0, 1)

integral of e^{5-x}

\int e^{5 - x} d x