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

Lösung

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

8 (- \frac{3}{4} arctan (\frac{1}{3} tan (\frac{x}{2})) + \frac{1}{2} arctan (tan (\frac{x}{2})) - \frac{1}{4} cos^{2} (\frac{x}{2}) tan (\frac{x}{2})) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{8 u ^{2}}{( u ^{2} + 3 ) ( u ^{2} + 1 ) ^{2}} d u

Entferne die Konstante:

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

= 8 \cdot \int \frac{u ^{2}}{( u ^{2} + 3 ) ( u ^{2} + 1 ) ^{2}} d u

Ermittle den Partialbruch von

\frac{u ^{2}}{( u ^{2} + 3 ) ( u ^{2} + 1 ) ^{2}} : - \frac{3}{4 ( u ^{2} + 3 )} + \frac{3}{4 ( u ^{2} + 1 )} - \frac{1}{2 ( u ^{2} + 1 ) ^{2}}

= 8 \cdot \int - \frac{3}{4 ( u ^{2} + 3 )} + \frac{3}{4 ( u ^{2} + 1 )} - \frac{1}{2 ( u ^{2} + 1 ) ^{2}} d u

Wende die Summenregel an:

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

= 8 (- \int \frac{3}{4 ( u ^{2} + 3 )} d u + \int \frac{3}{4 ( u ^{2} + 1 )} d u - \int \frac{1}{2 ( u ^{2} + 1 ) ^{2}} d u)

\int \frac{3}{4 ( u ^{2} + 3 )} d u = \frac{3}{4} arctan (\frac{u}{3})

\int \frac{3}{4 ( u ^{2} + 1 )} d u = \frac{3}{4} arctan (u)

\int \frac{1}{2 ( u ^{2} + 1 ) ^{2}} d u = \frac{1}{4} (arctan (u) + \frac{u}{1 + u ^{2}})

= 8 (- \frac{3}{4} arctan (\frac{u}{3}) + \frac{3}{4} arctan (u) - \frac{1}{4} (arctan (u) + \frac{u}{1 + u ^{2}}))

Setze in

u = tan (\frac{x}{2})

ein

= 8 (- \frac{3}{4} arctan (\frac{tan ( \frac{x}{2} )}{3}) + \frac{3}{4} arctan (tan (\frac{x}{2})) - \frac{1}{4} (arctan (tan (\frac{x}{2})) + \frac{tan ( \frac{x}{2} )}{1 + tan ^{2} ( \frac{x}{2} )}))

Vereinfache

8 (- \frac{3}{4} arctan (\frac{tan ( \frac{x}{2} )}{3}) + \frac{3}{4} arctan (tan (\frac{x}{2})) - \frac{1}{4} (arctan (tan (\frac{x}{2})) + \frac{tan ( \frac{x}{2} )}{1 + tan ^{2} ( \frac{x}{2} )})) : 8 (- \frac{3}{4} arctan (\frac{1}{3} tan (\frac{x}{2})) + \frac{1}{2} arctan (tan (\frac{x}{2})) - \frac{1}{4} cos^{2} (\frac{x}{2}) tan (\frac{x}{2}))

= 8 (- \frac{3}{4} arctan (\frac{1}{3} tan (\frac{x}{2})) + \frac{1}{2} arctan (tan (\frac{x}{2})) - \frac{1}{4} cos^{2} (\frac{x}{2}) tan (\frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= 8 (- \frac{3}{4} arctan (\frac{1}{3} tan (\frac{x}{2})) + \frac{1}{2} arctan (tan (\frac{x}{2})) - \frac{1}{4} cos^{2} (\frac{x}{2}) tan (\frac{x}{2})) + C

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