integral of (sin(2x))/(1+e^{cos^2(x))}

Lösung

\int \frac{sin ( 2 x )}{1 + e ^{c o s^{2} (x)}} d x

- ln (e^{c o s^{2} (x)}) + ln e^{c o s^{2} (x)} + 1 + C

Schritte zur Lösung

\int \frac{sin ( 2 x )}{1 + e ^{c o s^{2} (x)}} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u + 1} d u = ln ∣ u + 1 ∣

= - (ln ∣ u ∣ - ln ∣ u + 1 ∣)

Setze in

u = e^{c o s^{2} (x)}

ein

= - (ln e^{c o s^{2} (x)} - ln e^{c o s^{2} (x)} + 1)

Vereinfache

- (ln e^{c o s^{2} (x)} - ln e^{c o s^{2} (x)} + 1) : - ln (e^{c o s^{2} (x)}) + ln e^{c o s^{2} (x)} + 1

= - ln (e^{c o s^{2} (x)}) + ln e^{c o s^{2} (x)} + 1

Füge eine Konstante zur Lösung hinzu

= - ln (e^{c o s^{2} (x)}) + ln e^{c o s^{2} (x)} + 1 + C

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

\int \frac{8 x}{x ^{2} + 4} d x

\int \frac{6 x}{x ^{2} + 3} d x

\int \frac{1}{5 - 16 x ^{2}} d x

\int 7 x (2 x + 1)^{- 3} d x