integral of (sin(a)x)/(sqrt(c+cos(a)x))

Lösung

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

sec^{2} (a) sin (a) (\frac{2}{3} (c + x cos (a))^{\frac{3}{2}} - 2 c c + x cos (a)) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

= sin (a) \cdot \int \frac{x}{c + cos ( a ) x} d x

Wende U-Substitution an

= sin (a) \cdot \int \frac{u - c}{u cos ^{2} ( a )} d u

Entferne die Konstante:

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

= sin (a) \frac{1}{cos ^{2} ( a )} \cdot \int \frac{u - c}{u} d u

Multipliziere aus

\frac{u - c}{u} : u - \frac{c}{u}

= sin (a) \frac{1}{cos ^{2} ( a )} \cdot \int u - \frac{c}{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

= sin (a) \frac{1}{cos ^{2} ( a )} (\int u d u - \int \frac{c}{u} d u)

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

\int \frac{c}{u} d u = 2 c u

= sin (a) \frac{1}{cos ^{2} ( a )} (\frac{2}{3} u^{\frac{3}{2}} - 2 c u)

Setze in

u = c + cos (a) x

ein

= sin (a) \frac{1}{cos ^{2} ( a )} (\frac{2}{3} (c + cos (a) x)^{\frac{3}{2}} - 2 c c + cos (a) x)

Verwende die grundlegende trigonometrische Identität:

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

= sec^{2} (a) sin (a) (\frac{2}{3} (c + x cos (a))^{\frac{3}{2}} - 2 c c + x cos (a))

Füge eine Konstante zur Lösung hinzu

= sec^{2} (a) sin (a) (\frac{2}{3} (c + x cos (a))^{\frac{3}{2}} - 2 c c + x cos (a)) + C

\int \frac{1}{7} x^{- \frac{6}{7}} d x

t an g e n t y^{2} + x y + x^{2} = 1, (0, 1)

x \frac{d y}{d x} = x^{- 3} - 2 y

\int_{1}^{4} \frac{3}{x} d x

in v erse l a pl a ce \frac{3}{2 s ^{2} + 3 s + 1}