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Beliebt Rechnen >

integral of x^3cos^2(x)

Lösung

\int x^{3} cos^{2} (x) d x

Lösung

\frac{1}{4} x^{4} cos^{2} (x) + \frac{1}{16} (- 2 x^{4} cos (2 x) + 4 x^{3} sin (2 x) - 3 (- 2 x^{2} cos (2 x) + 2 x sin (2 x) + cos (2 x))) + C

Schritte zur Lösung

\int x^{3} cos^{2} (x) d x

Wende die partielle Integration an

= \frac{1}{4} x^{4} cos^{2} (x) - \int - \frac{1}{4} x^{4} sin (2 x) d x

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} x^{4} cos^{2} (x) + \frac{1}{16} (- 2 x^{4} cos (2 x) + 4 x^{3} sin (2 x) - 3 (- 2 x^{2} cos (2 x) + 2 x sin (2 x) + cos (2 x))) + C

Graph

Beliebte Beispiele

integral of 1/(sqrt(4^2-x^2))

\int \frac{1}{4 ^{2} - x ^{2}} d x

derivative t/((t-6)^2)

d er i v a t i v e \frac{t}{( t - 6 ) ^{2}}

limit as x approaches-8-of (|x+8|)/(x+8)

x \to - 8 - lim (\frac{∣ x + 8 ∣}{x + 8})

sum from n=0 to infinity of (x^n)/(n+1)

n = 0 \sum \infty \frac{x ^{n}}{n + 1}

derivative f(x)=(3x+1)(5x-4)

d er i v a t i v e f (x) = (3 x + 1) (5 x - 4)