integral of 1/(sqrt(cos(x)-cos^2(x)))

Lösung

\int \frac{1}{cos ( x ) - cos ^{2} ( x )} d x

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

Schritte zur Lösung

\int \frac{1}{cos ( x ) - cos ^{2} ( x )} d x

Wende U-Substitution an

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

Wende U-Substitution an

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{\partial}{\partial y} (- 6 x y - x^{6} - y^{6})

d er i v a t i v e \frac{( 4 x + 3 ) ^{8}}{( x ) ^{4}}

x \to 0 lim (\frac{6}{1 + x})

\frac{\partial}{\partial x} (e^{5 x + 3 y})

\frac{d}{d y} (\frac{1}{t} + \frac{1}{t ^{2}} - \frac{y}{t ^{2} + y ^{2}})