Analysis Beispiele

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

$\int [\frac{\sin (x)}{\cos^{2} (x)} + x] d x$

Schritt 1

Entferne die Klammern.

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

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

Schritt 2

Zerlege das einzelne Integral in mehrere Integrale.

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

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

Schritt 3

Multipliziere mit

1

$1$ .

\int \frac{\sin (x) \cdot 1}{\cos^{2} (x)} d x + \int x d x

$\int \frac{\sin (x) \cdot 1}{\cos^{2} (x)} d x + \int x d x$

Schritt 4

Faktorisiere

\cos (x)

$\cos (x)$ aus

\cos^{2} (x)

$\cos^{2} (x)$ heraus.

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

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

Schritt 5

Separiere Brüche.

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

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

Schritt 6

Wandle von

\frac{\sin (x)}{\cos (x)}

$\frac{\sin (x)}{\cos (x)}$ nach

\tan (x)

$\tan (x)$ um.

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

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

Schritt 7

Wandle von

\frac{1}{\cos (x)}

$\frac{1}{\cos (x)}$ nach

\sec (x)

$\sec (x)$ um.

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

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

Schritt 8

Da die Ableitung von

\sec (x)

$\sec (x)$ gleich

\tan (x) \sec (x)

$\tan (x) \sec (x)$ ist, ist das Integral von

\tan (x) \sec (x)

$\tan (x) \sec (x)$ gleich

\sec (x)

$\sec (x)$ .

\sec (x) + C + \int x d x

$\sec (x) + C + \int x d x$

Schritt 9

Gemäß der Potenzregel ist das Integral von

x

$x$ nach

x

$x$ gleich

\frac{1}{2} x^{2}

$\frac{1}{2} x^{2}$ .

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

$\sec (x) + C + \frac{1}{2} x^{2} + C$

Schritt 10

Vereinfache.

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

$\sec (x) + \frac{1}{2} x^{2} + C$