integral of cos(x)sqrt(4+sin^2(x))

Lösung

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

4 (\frac{1}{8} sin (x) 4 + sin^{2} (x) + \frac{1}{2} ln (\frac{1}{2} sin (x) + 4 + sin^{2} (x))) + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int cos (x) - cos^{2} (x) + 5 d x

Trigonometrische Substitution anwenden

= \int u^{2} + 4 d u

Trigonometrische Substitution anwenden

= \int 4 sec^{3} (v) d v

Entferne die Konstante:

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

= 4 \cdot \int sec^{3} (v) d v

Wende integrale Reduktion an

= 4 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

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

= 4 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Ersetze zurück

= 4 (\frac{sec ^{2} ( arctan ( \frac{1}{2} sin ( x ) ) ) sin ( arctan ( \frac{1}{2} sin ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{2} sin (x))) + sec (arctan (\frac{1}{2} sin (x))))

Vereinfache

4 (\frac{sec ^{2} ( arctan ( \frac{1}{2} sin ( x ) ) ) sin ( arctan ( \frac{1}{2} sin ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{2} sin (x))) + sec (arctan (\frac{1}{2} sin (x)))) : 4 (\frac{1}{8} sin (x) 4 + sin^{2} (x) + \frac{1}{2} ln (\frac{1}{2} sin (x) + 4 + sin^{2} (x)))

= 4 (\frac{1}{8} sin (x) 4 + sin^{2} (x) + \frac{1}{2} ln (\frac{1}{2} sin (x) + 4 + sin^{2} (x)))

Füge eine Konstante zur Lösung hinzu

= 4 (\frac{1}{8} sin (x) 4 + sin^{2} (x) + \frac{1}{2} ln (\frac{1}{2} sin (x) + 4 + sin^{2} (x))) + C

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

\int \frac{1 - x ln ( x )}{x e ^{x}} d x

\int 1 - x - y d y

\int 6 sec^{2} (x) - 9 x^{3} d x

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