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integral of csc^3(x)*sec(x)

Lösung

\int csc^{3} (x) \cdot sec (x) d x

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

Schritte zur Lösung

\int csc^{3} (x) sec (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \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 (x^{2} + a x - 3)^{2} d x

\int_{0}^{1} 2 x^{2} - 1 d x

\frac{d ^{2}}{d x ^{2}} ((x + 1) e^{x})

n = 1 \sum \infty sin (πn)

\frac{d}{d z} (\frac{i}{2 i} ln (\frac{i + i z}{i - i z}))