integral of (sqrt(1+cos(x)))/(sin(x))

Lösung

\int \frac{1 + cos ( x )}{sin ( x )} d x

- 2 (\frac{1}{2} ln sec (\frac{x}{2}) + 1 - \frac{1}{2} ln sec (\frac{x}{2}) - 1) + C

Schritte zur Lösung

\int \frac{1 + cos ( x )}{sin ( x )} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Faktorisiere

v^{2} - 1 : - (- v^{2} + 1)

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

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

= 2 (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Ersetze zurück

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

Vereinfache

2 - \frac{ln 1 + tan ^{2} ( \frac{x}{2} ) + 1}{2} - \frac{ln 1 + tan ^{2} ( \frac{x}{2} ) - 1}{2} : - 2 (\frac{1}{2} ln sec (\frac{x}{2}) + 1 - \frac{1}{2} ln sec (\frac{x}{2}) - 1)

= - 2 (\frac{1}{2} ln sec (\frac{x}{2}) + 1 - \frac{1}{2} ln sec (\frac{x}{2}) - 1)

Füge eine Konstante zur Lösung hinzu

= - 2 (\frac{1}{2} ln sec (\frac{x}{2}) + 1 - \frac{1}{2} ln sec (\frac{x}{2}) - 1) + C

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

\int 4 x sec^{2} (x) - 4 x tan^{2} (x) d x

\int \frac{1}{4 + 49 x ^{2}} d x

\int \frac{e ^{x}}{e ^{2 x}} d x

\int \frac{1}{( x ^{2} - 6 x + 10 ) ^{\frac{5}{2}}} d x