English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of [(a^3)/(x^2+a^2)]^2

Lösung

\int [\frac{a ^{3}}{x ^{2} + a ^{2}}]^{2} d x

\frac{a ^{3}}{4} (2 arctan (\frac{x}{a}) + sin (2 arctan (\frac{x}{a}))) + C

Schritte zur Lösung

\int [\frac{a ^{3}}{x ^{2} + a ^{2}}]^{2} d x

Vereinfache

[\frac{a ^{3}}{x ^{2} + a ^{2}}]^{2} : \frac{a ^{6}}{( x ^{2} + a ^{2} ) ^{2}}

= \int \frac{a ^{6}}{( x ^{2} + a ^{2} ) ^{2}} d x

Entferne die Konstante:

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

= a^{6} \cdot \int \frac{1}{( x ^{2} + a ^{2} ) ^{2}} d x

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

= a^{6} \cdot \int \frac{1}{( a ^{2} ( \frac{x ^{2}}{a ^{2}} + 1 ) ) ^{2}} d x

Wende integrale Substitution an

= a^{6} \cdot \int \frac{1}{a ^{3} ( u ^{2} + 1 ) ^{2}} d u

Entferne die Konstante:

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

= a^{6} \frac{1}{a ^{3}} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u

Trigonometrische Substitution anwenden

= a^{6} \frac{1}{a ^{3}} \cdot \int \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= a^{6} \frac{1}{a ^{3}} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Entferne die Konstante:

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

= a^{6} \frac{1}{a ^{3}} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) d v

Wende die Summenregel an:

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

= a^{6} \frac{1}{a ^{3}} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= a^{6} \frac{1}{a ^{3}} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Ersetze zurück

= a^{6} \frac{1}{a ^{3}} \cdot \frac{1}{2} (arctan (\frac{x}{a}) + \frac{1}{2} sin (2 arctan (\frac{x}{a})))

Vereinfache

a^{6} \frac{1}{a ^{3}} \cdot \frac{1}{2} (arctan (\frac{x}{a}) + \frac{1}{2} sin (2 arctan (\frac{x}{a}))) : \frac{a ^{3}}{4} (2 arctan (\frac{x}{a}) + sin (2 arctan (\frac{x}{a})))

= \frac{a ^{3}}{4} (2 arctan (\frac{x}{a}) + sin (2 arctan (\frac{x}{a})))

Füge eine Konstante zur Lösung hinzu

= \frac{a ^{3}}{4} (2 arctan (\frac{x}{a}) + sin (2 arctan (\frac{x}{a}))) + C

n = 1 \sum \infty (\frac{n}{n + 4})^{n}

\int cos (4 x - 3) d x

a re a y = 3 x - x^{2}, y = - 3 x

s l o p e (\frac{2}{7}, 5), (10, 39)

\int_{15}^{35} (x^{2} x) d x