integral of cos^4(xta)n^3x

Lösung

\int cos^{4} (x t a) n^{3} x d x

\frac{n ^{3}}{128 a ^{2} t ^{2}} (24 a^{2} t^{2} x^{2} + 128 a t x cos^{3} (a t x) sin (a t x) - 12 a t x sin (4 a t x) + 32 cos^{4} (a t x) - 3 cos (4 a t x)) + C

Schritte zur Lösung

\int cos^{4} (x t a) n^{3} x d x

Entferne die Konstante:

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

= n^{3} \cdot \int cos^{4} (a t x) x d x

Wende U-Substitution an

= n^{3} \cdot \int \frac{u cos ^{4} ( u )}{a ^{2} t ^{2}} d u

Entferne die Konstante:

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

= n^{3} \frac{1}{a ^{2} t ^{2}} \cdot \int u cos^{4} (u) d u

Wende die partielle Integration an

= n^{3} \frac{1}{a ^{2} t ^{2}} (u (cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u))) - \int cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u)) d u)

\int cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u)) d u = - \frac{cos ^{4} ( u )}{4} + \frac{3}{8} (\frac{u ^{2}}{2} + \frac{1}{16} cos (4 u))

= n^{3} \frac{1}{a ^{2} t ^{2}} (u (cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u))) - (- \frac{cos ^{4} ( u )}{4} + \frac{3}{8} (\frac{u ^{2}}{2} + \frac{1}{16} cos (4 u))))

Setze in

u = a t x

ein

= n^{3} \frac{1}{a ^{2} t ^{2}} (a t x (cos^{3} (a t x) sin (a t x) + \frac{3}{8} (a t x - \frac{1}{4} sin (4 a t x))) - (- \frac{cos ^{4} ( a t x )}{4} + \frac{3}{8} (\frac{( a t x ) ^{2}}{2} + \frac{1}{16} cos (4 a t x))))

Vereinfache

n^{3} \frac{1}{a ^{2} t ^{2}} (a t x (cos^{3} (a t x) sin (a t x) + \frac{3}{8} (a t x - \frac{1}{4} sin (4 a t x))) - (- \frac{cos ^{4} ( a t x )}{4} + \frac{3}{8} (\frac{( a t x ) ^{2}}{2} + \frac{1}{16} cos (4 a t x)))) : \frac{n ^{3}}{128 a ^{2} t ^{2}} (24 a^{2} t^{2} x^{2} + 128 a t x cos^{3} (a t x) sin (a t x) - 12 a t x sin (4 a t x) + 32 cos^{4} (a t x) - 3 cos (4 a t x))

= \frac{n ^{3}}{128 a ^{2} t ^{2}} (24 a^{2} t^{2} x^{2} + 128 a t x cos^{3} (a t x) sin (a t x) - 12 a t x sin (4 a t x) + 32 cos^{4} (a t x) - 3 cos (4 a t x))

Füge eine Konstante zur Lösung hinzu

= \frac{n ^{3}}{128 a ^{2} t ^{2}} (24 a^{2} t^{2} x^{2} + 128 a t x cos^{3} (a t x) sin (a t x) - 12 a t x sin (4 a t x) + 32 cos^{4} (a t x) - 3 cos (4 a t x)) + C

t a y l or e^{- h}

s l o p e (- 2, - 7), (2, - 1)

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\frac{d}{d y} (\frac{1}{y ( 1 - x y )})

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