integral of 1/(sin(x)+tan(x))

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

\frac{( u + 1 ) ( u - 1 )}{u} : u - \frac{1}{u}

= - \frac{1}{2} \cdot \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

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

Setze in

u = tan (\frac{x}{2})

ein

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (i sin (x))

y^{^{''}} = (y^{^{'}})^{2}

\frac{d}{d x} (ln (x^{\frac{1}{3}}))

\frac{d}{d x} (cos (x (1 + sin (x))^{10}))

f (x) = 2 cos (x^{3}) - \frac{2}{x}