Analysis Beispiele

{sec}^{3} (x)

$\sec^{3} (x)$

Schritt 1

Faktorisiere

sec (x)

$\sec (x)$ aus

{sec}^{3} (x)

$\sec^{3} (x)$ heraus.

\int sec (x) {sec}^{2} (x) d x

$\int \sec (x) \sec^{2} (x) d x$

Schritt 2

Integriere partiell durch Anwendung der Formel

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , mit

u = sec (x)

$u = \sec (x)$ und

d v = {sec}^{2} (x)

$d v = \sec^{2} (x)$ .

sec (x) tan (x) - \int tan (x) (sec (x) tan (x)) d x

$\sec (x) \tan (x) - \int \tan (x) (\sec (x) \tan (x)) d x$

Schritt 3

Potenziere

tan (x)

$\tan (x)$ mit

1

$1$ .

sec (x) tan (x) - \int {tan}^{1} (x) tan (x) sec (x) d x

$\sec (x) \tan (x) - \int \tan^{1} (x) \tan (x) \sec (x) d x$

Schritt 4

Potenziere

tan (x)

$\tan (x)$ mit

1

$1$ .

sec (x) tan (x) - \int {tan}^{1} (x) {tan}^{1} (x) sec (x) d x

$\sec (x) \tan (x) - \int \tan^{1} (x) \tan^{1} (x) \sec (x) d x$

Schritt 5

Wende die Exponentenregel

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

sec (x) tan (x) - \int {tan (x)}^{1 + 1} sec (x) d x

$\sec (x) \tan (x) - \int {\tan (x)}^{1 + 1} \sec (x) d x$

Schritt 6

Vereinfache den Ausdruck.

Tippen, um mehr Schritte zu sehen ...

Schritt 6.1

Addiere

1

$1$ und

1

$1$ .

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

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

Schritt 6.2

Stelle

{tan}^{2} (x)

$\tan^{2} (x)$ und

sec (x)

$\sec (x)$ um.

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

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

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

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

Schritt 7

Schreibe

{tan}^{2} (x)

$\tan^{2} (x)$ in

- 1 + {sec}^{2} (x)

$- 1 + \sec^{2} (x)$ um unter Verwendung des trigonometrischen Pythagoras.

sec (x) tan (x) - \int sec (x) (- 1 + {sec}^{2} (x)) d x

$\sec (x) \tan (x) - \int \sec (x) (- 1 + \sec^{2} (x)) d x$

Schritt 8

Vereinfache durch Ausmultiplizieren.

Tippen, um mehr Schritte zu sehen ...

Schritt 8.1

Schreibe die Potenz um als ein Produkt.

sec (x) tan (x) - \int sec (x) (- 1 + sec (x) sec (x)) d x

$\sec (x) \tan (x) - \int \sec (x) (- 1 + \sec (x) \sec (x)) d x$

Schritt 8.2

Wende das Distributivgesetz an.

sec (x) tan (x) - \int sec (x) \cdot - 1 + sec (x) (sec (x) sec (x)) d x

$\sec (x) \tan (x) - \int \sec (x) \cdot - 1 + \sec (x) (\sec (x) \sec (x)) d x$

Schritt 8.3

Stelle

sec (x)

$\sec (x)$ und

- 1

$- 1$ um.

sec (x) tan (x) - \int - 1 \cdot sec (x) + sec (x) (sec (x) sec (x)) d x

$\sec (x) \tan (x) - \int - 1 \cdot \sec (x) + \sec (x) (\sec (x) \sec (x)) d x$

sec (x) tan (x) - \int - 1 \cdot sec (x) + sec (x) (sec (x) sec (x)) d x

$\sec (x) \tan (x) - \int - 1 \cdot \sec (x) + \sec (x) (\sec (x) \sec (x)) d x$

Schritt 9

Potenziere

sec (x)

$\sec (x)$ mit

1

$1$ .

sec (x) tan (x) - \int - 1 sec (x) + {sec}^{1} (x) sec (x) sec (x) d x

$\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{1} (x) \sec (x) \sec (x) d x$

Schritt 10

Potenziere

sec (x)

$\sec (x)$ mit

1

$1$ .

sec (x) tan (x) - \int - 1 sec (x) + {sec}^{1} (x) {sec}^{1} (x) sec (x) d x

$\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{1} (x) \sec^{1} (x) \sec (x) d x$

Schritt 11

Wende die Exponentenregel

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

sec (x) tan (x) - \int - 1 sec (x) + {sec (x)}^{1 + 1} sec (x) d x

$\sec (x) \tan (x) - \int - 1 \sec (x) + {\sec (x)}^{1 + 1} \sec (x) d x$

Schritt 12

Addiere

1

$1$ und

1

$1$ .

sec (x) tan (x) - \int - 1 sec (x) + {sec}^{2} (x) sec (x) d x

$\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{2} (x) \sec (x) d x$

Schritt 13

Potenziere

sec (x)

$\sec (x)$ mit

1

$1$ .

sec (x) tan (x) - \int - 1 sec (x) + {sec}^{2} (x) {sec}^{1} (x) d x

$\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{2} (x) \sec^{1} (x) d x$

Schritt 14

Wende die Exponentenregel

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

sec (x) tan (x) - \int - 1 sec (x) + {sec (x)}^{2 + 1} d x

$\sec (x) \tan (x) - \int - 1 \sec (x) + {\sec (x)}^{2 + 1} d x$

Schritt 15

Addiere

2

$2$ und

1

$1$ .

sec (x) tan (x) - \int - 1 sec (x) + {sec}^{3} (x) d x

$\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{3} (x) d x$

Schritt 16

Zerlege das einzelne Integral in mehrere Integrale.

sec (x) tan (x) - (\int - 1 sec (x) d x + \int {sec}^{3} (x) d x)

$\sec (x) \tan (x) - (\int - 1 \sec (x) d x + \int \sec^{3} (x) d x)$

Schritt 17

- 1

$- 1$ konstant bezüglich

x

$x$ ist, ziehe

- 1

$- 1$ aus dem Integral.

sec (x) tan (x) - (- \int sec (x) d x + \int {sec}^{3} (x) d x)

$\sec (x) \tan (x) - (- \int \sec (x) d x + \int \sec^{3} (x) d x)$

Schritt 18

Das Integral von

sec (x)

$\sec (x)$ nach

x

$x$ ist

ln (| sec (x) + tan (x) |)

$\ln (| \sec (x) + \tan (x) |)$ .

sec (x) tan (x) - (- (ln (| sec (x) + tan (x) |) + C) + \int {sec}^{3} (x) d x)

$\sec (x) \tan (x) - (- (\ln (| \sec (x) + \tan (x) |) + C) + \int \sec^{3} (x) d x)$

Schritt 19

Vereinfache durch Ausmultiplizieren.

Tippen, um mehr Schritte zu sehen ...

Schritt 19.1

Wende das Distributivgesetz an.

sec (x) tan (x) - - (ln (| sec (x) + tan (x) |) + C) - \int {sec}^{3} (x) d x

$\sec (x) \tan (x) - - (\ln (| \sec (x) + \tan (x) |) + C) - \int \sec^{3} (x) d x$

Schritt 19.2

Mutltipliziere

- 1

$- 1$ mit

- 1

$- 1$ .

sec (x) tan (x) + 1 (ln (| sec (x) + tan (x) |) + C) - \int {sec}^{3} (x) d x

$\sec (x) \tan (x) + 1 (\ln (| \sec (x) + \tan (x) |) + C) - \int \sec^{3} (x) d x$

sec (x) tan (x) + 1 (ln (| sec (x) + tan (x) |) + C) - \int {sec}^{3} (x) d x

$\sec (x) \tan (x) + 1 (\ln (| \sec (x) + \tan (x) |) + C) - \int \sec^{3} (x) d x$

Schritt 20

Wenn nach

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

$\int \sec^{3} (x) d x$ aufgelöst wird, erhalten wir

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

$\int \sec^{3} (x) d x$ =

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

$\frac{\sec (x) \tan (x) + 1 (\ln (| \sec (x) + \tan (x) |) + C)}{2}$ .

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

$\frac{\sec (x) \tan (x) + 1 (\ln (| \sec (x) + \tan (x) |) + C)}{2} + C$

Schritt 21

Mutltipliziere

ln (| sec (x) + tan (x) |) + C

$\ln (| \sec (x) + \tan (x) |) + C$ mit

$1$ .

$\frac{\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |) + C}{2} + C$

Schritt 22

Vereinfache.

$\frac{1}{2} (\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |)) + C$