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Beliebt Rechnen >

integral of (sqrt(x^2+1))

Lösung

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

Lösung

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

Schritte zur Lösung

\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

(d^2)/(dx^2)(6x^2+5x)

\frac{d ^{2}}{d x ^{2}} (6 x^{2} + 5 x)

(t+1)dt-1/(y^2)dy=0

(t + 1) d t - \frac{1}{y ^{2}} d y = 0

integral of (2w-5)/(w^2-3w+9/2)

\int \frac{2 w - 5}{w ^{2} - 3 w + \frac{9}{2}} d x

fläche 2x+y^2=8,x=y

a re a 2 x + y^{2} = 8, x = y

(\partial)/(\partial x)(6xy-2y^2)

\frac{\partial}{\partial x} (6 x y - 2 y^{2})