de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (sin^2(3x)+cos(3x))^2

Lösung

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

Lösung

\frac{7}{8} x + \frac{2}{9} sin^{3} (3 x) - \frac{1}{12} sin^{3} (3 x) cos (3 x) + \frac{1}{48} sin (6 x) + C

Schritte zur Lösung

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

Multipliziere aus

(sin^{2} (3 x) + cos (3 x))^{2} : sin^{4} (3 x) + 2 sin^{2} (3 x) cos (3 x) + cos^{2} (3 x)

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int sin^{4} (3 x) d x = - \frac{1}{12} sin^{3} (3 x) cos (3 x) + \frac{1}{16} (6 x - sin (6 x))

\int 2 sin^{2} (3 x) cos (3 x) d x = \frac{2}{9} sin^{3} (3 x)

\int cos^{2} (3 x) d x = \frac{1}{2} (x + \frac{1}{6} sin (6 x))

= - \frac{1}{12} sin^{3} (3 x) cos (3 x) + \frac{1}{16} (6 x - sin (6 x)) + \frac{2}{9} sin^{3} (3 x) + \frac{1}{2} (x + \frac{1}{6} sin (6 x))

Vereinfache

- \frac{1}{12} sin^{3} (3 x) cos (3 x) + \frac{1}{16} (6 x - sin (6 x)) + \frac{2}{9} sin^{3} (3 x) + \frac{1}{2} (x + \frac{1}{6} sin (6 x)) : \frac{7}{8} x + \frac{2}{9} sin^{3} (3 x) - \frac{1}{12} sin^{3} (3 x) cos (3 x) + \frac{1}{48} sin (6 x)

= \frac{7}{8} x + \frac{2}{9} sin^{3} (3 x) - \frac{1}{12} sin^{3} (3 x) cos (3 x) + \frac{1}{48} sin (6 x)

Füge eine Konstante zur Lösung hinzu

= \frac{7}{8} x + \frac{2}{9} sin^{3} (3 x) - \frac{1}{12} sin^{3} (3 x) cos (3 x) + \frac{1}{48} sin (6 x) + C

Graph

Beliebte Beispiele

integral of (5× 2^x)/(3^x)

\int \frac{5 \times 2 ^{x}}{3 ^{x}} d x

integral of 1/(x(x+1)(x^2+x+1))

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

integral of (1-sin(2x))^2

\int (1 - sin (2 x))^{2} d x

integral of (2t-1)/((t+1)^3)

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

integral of 2xsqrt(a^2-x^2)

\int 2 x a^{2} - x^{2} d x