integral of (sin(x)+tan(x))/(cos(x))

Lösung

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

- (ln ∣ cos (x) ∣ - sec (x)) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= - (ln ∣ u ∣ - \frac{1}{u})

Setze in

u = cos (x)

ein

= - (ln ∣ cos (x) ∣ - \frac{1}{cos ( x )})

Verwende die grundlegende trigonometrische Identität:

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

= - (ln ∣ cos (x) ∣ - sec (x))

Füge eine Konstante zur Lösung hinzu

= - (ln ∣ cos (x) ∣ - sec (x)) + C

\int \frac{17 x}{x ^{4} + 1} d x

\int (3 x^{3} + 2)^{2} d x

\int (x - \frac{1}{x ^{2}}) (x^{2} + \frac{2}{x})^{5} d x

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

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