integral of 4sin^2(5x+3)+10x^{2/3}-1

Lösung

\int 4 sin^{2} (5 x + 3) + 10 x^{\frac{2}{3}} - 1 d x

3 x - 4 x cos^{2} (5 x + 3) + 2 x cos (2 (5 x + 3)) + 6 x^{\frac{5}{3}} - \frac{1}{5} sin (2 (5 x + 3)) + C

Schritte zur Lösung

\int 4 sin^{2} (5 x + 3) + 10 x^{\frac{2}{3}} - 1 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 4 sin^{2} (5 x + 3) d x + \int 10 x^{\frac{2}{3}} d x - \int 1 d x

\int 4 sin^{2} (5 x + 3) d x = 2 (2 x - 2 (x cos^{2} (5 x + 3) + \frac{1}{20} (- 10 x cos (2 (5 x + 3)) + sin (2 (5 x + 3)))))

\int 10 x^{\frac{2}{3}} d x = 6 x^{\frac{5}{3}}

\int 1 d x = x

= 2 (2 x - 2 (x cos^{2} (5 x + 3) + \frac{1}{20} (- 10 x cos (2 (5 x + 3)) + sin (2 (5 x + 3))))) + 6 x^{\frac{5}{3}} - x

Vereinfache

2 (2 x - 2 (x cos^{2} (5 x + 3) + \frac{1}{20} (- 10 x cos (2 (5 x + 3)) + sin (2 (5 x + 3))))) + 6 x^{\frac{5}{3}} - x : 3 x - 4 x cos^{2} (5 x + 3) + 2 x cos (2 (5 x + 3)) + 6 x^{\frac{5}{3}} - \frac{1}{5} sin (2 (5 x + 3))

= 3 x - 4 x cos^{2} (5 x + 3) + 2 x cos (2 (5 x + 3)) + 6 x^{\frac{5}{3}} - \frac{1}{5} sin (2 (5 x + 3))

Füge eine Konstante zur Lösung hinzu

= 3 x - 4 x cos^{2} (5 x + 3) + 2 x cos (2 (5 x + 3)) + 6 x^{\frac{5}{3}} - \frac{1}{5} sin (2 (5 x + 3)) + C

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

e x p an d 6 x (x^{2} - 5)^{2}

\frac{d}{d x} (\frac{9 x}{x + 4})

\frac{d y}{d x} = 4 y^{2}

s l o p e f (x) = 3 x - x^{2}