integral of [cos(x/2)-sin(3 x/2)]

Lösung

\int [cos (\frac{x}{2}) - sin (3 \frac{x}{2})] d x

2 (sin (\frac{x}{2}) + \frac{1}{3} cos (\frac{3 x}{2})) + C

Schritte zur Lösung

\int cos (\frac{x}{2}) - sin (3 \cdot \frac{x}{2}) d x

Wende U-Substitution an

= \int (cos (u) - sin (3 u)) \cdot 2 d u

Entferne die Konstante:

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

= 2 \cdot \int cos (u) - sin (3 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

= 2 (\int cos (u) d u - \int sin (3 u) d u)

\int cos (u) d u = sin (u)

\int sin (3 u) d u = - \frac{1}{3} cos (3 u)

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

Setze in

u = \frac{x}{2}

ein

= 2 (sin (\frac{x}{2}) - (- \frac{1}{3} cos (3 \cdot \frac{x}{2})))

Vereinfache

2 (sin (\frac{x}{2}) - (- \frac{1}{3} cos (3 \cdot \frac{x}{2}))) : 2 (sin (\frac{x}{2}) + \frac{1}{3} cos (\frac{3 x}{2}))

= 2 (sin (\frac{x}{2}) + \frac{1}{3} cos (\frac{3 x}{2}))

Füge eine Konstante zur Lösung hinzu

= 2 (sin (\frac{x}{2}) + \frac{1}{3} cos (\frac{3 x}{2})) + C

\frac{d}{d x} (\frac{7}{x ^{- 2}} + \frac{5}{x ^{- 3}})

a re a x, \frac{1}{3} x, x = 25

x \to 0 lim (\frac{100}{2})

\frac{d}{d x} (tan^{4} (3) x^{2})

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