de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of y(x^2+1)^{1/2}

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

= y \cdot \int (1 + x^{2})^{\frac{1}{2}} d x

Trigonometrische Substitution anwenden

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

Wende integrale Reduktion an

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

Setze in

u = arctan (x)

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of sqrt(3r^2)

\int 3 r^{2} d r

integral of x^7sin(2x^4)

\int x^{7} sin (2 x^{4}) d x

integral of tan^4(x)cos^3(x)

\int tan^{4} (x) cos^{3} (x) d x

integral of 1/((4x+1)^7)

\int \frac{1}{( 4 x + 1 ) ^{7}} d x

integral of x*(-2)/(sqrt(1-4x^2))

\int x \cdot \frac{- 2}{1 - 4 x ^{2}} d x