integral of 1/(sin^3(x)cos(x))

Lösung

\int \frac{1}{sin ^{3} ( x ) cos ( x )} d x

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

Schritte zur Lösung

\int \frac{1}{sin ^{3} ( x ) cos ( x )} d x

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Vereinfache

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

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

Multipliziere aus

\frac{u ^{2} + 1}{u} : u + \frac{1}{u}

= - \int u + \frac{1}{u} 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

= - (\int u d u + \int \frac{1}{u} d u)

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

\int \frac{1}{u} d u = ln ∣ u ∣

= - (\frac{u ^{2}}{2} + ln ∣ u ∣)

Setze in

u = cot (x)

ein

= - (\frac{cot ^{2} ( x )}{2} + ln ∣ cot (x) ∣)

Vereinfache

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

= - \frac{1}{2} cot^{2} (x) - ln ∣ cot (x) ∣

Füge eine Konstante zur Lösung hinzu

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

\int \frac{3 x ^{3}}{1 - x ^{4}} d x

\int \frac{5 - 2 x}{5 + 4 x - x ^{2}} d x

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

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

\int \frac{1}{1 + x - x ^{2}} d x