integral of x(sec^2(3x))

Lösung

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

\frac{1}{2} x^{2} sec^{2} (3 x) - \frac{1}{18} (9 x^{2} tan^{2} (3 x) - 6 x tan (3 x) - 2 ln ∣ cos (3 x) ∣ + 9 x^{2}) + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int x (1 + tan^{2} (3 x)) d x

Wende die partielle Integration an

= \frac{1}{2} x^{2} sec^{2} (3 x) - \int 3 x^{2} sec^{2} (3 x) tan (3 x) d x

\int 3 x^{2} sec^{2} (3 x) tan (3 x) d x = \frac{1}{18} (9 x^{2} tan^{2} (3 x) - 6 x tan (3 x) - 2 ln ∣ cos (3 x) ∣ + 9 x^{2})

= \frac{1}{2} x^{2} sec^{2} (3 x) - \frac{1}{18} (9 x^{2} tan^{2} (3 x) - 6 x tan (3 x) - 2 ln ∣ cos (3 x) ∣ + 9 x^{2})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} sec^{2} (3 x) - \frac{1}{18} (9 x^{2} tan^{2} (3 x) - 6 x tan (3 x) - 2 ln ∣ cos (3 x) ∣ + 9 x^{2}) + C

\int sin (6 x) sin (3 x) d x

\int \frac{2 x ^{2} + 3 x + 21}{( x + 1 ) ( x ^{2} + 9 )} d x

\int sec^{5} (y) tan^{3} (y) d y

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

\int \frac{1}{( y ^{2} + a ^{2} ) ^{\frac{3}{2}}} d y