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

Lösung

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

- \frac{1}{2} arctan (\frac{cos ( x )}{2}) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int - \frac{1}{4 + u ^{2}} d u

Entferne die Konstante:

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

= - \int \frac{1}{4 + u ^{2}} d u

Wende integrale Substitution an

= - \int \frac{1}{2 ( v ^{2} + 1 )} d v

Entferne die Konstante:

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

= - \frac{1}{2} \cdot \int \frac{1}{v ^{2} + 1} d v

Nutze das gemeinsame Integral :

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= - \frac{1}{2} arctan (v)

Ersetze zurück

= - \frac{1}{2} arctan (\frac{cos ( x )}{2})

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} arctan (\frac{cos ( x )}{2}) + C

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

\int_{2}^{4} 8 x d x

\int (2 sin (x) cos (x)) d x

\int 7 tan^{5} (x) d x

p a r i t y f (x) = x + sin (x)