integral of 64xsec^2(4x)

Lösung

\int 64 x sec^{2} (4 x) d x

64 (\frac{1}{2} x^{2} sec^{2} (4 x) - \frac{1}{16} (8 x^{2} tan^{2} (4 x) - 4 x tan (4 x) - ln ∣ cos (4 x) ∣ + 8 x^{2})) + C

Schritte zur Lösung

\int 64 x sec^{2} (4 x) d x

Entferne die Konstante:

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

= 64 \cdot \int x sec^{2} (4 x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 64 \cdot \int x (1 + tan^{2} (4 x)) d x

Wende die partielle Integration an

= 64 (\frac{1}{2} x^{2} sec^{2} (4 x) - \int 4 x^{2} sec^{2} (4 x) tan (4 x) d x)

\int 4 x^{2} sec^{2} (4 x) tan (4 x) d x = \frac{1}{16} (8 x^{2} tan^{2} (4 x) - 4 x tan (4 x) - ln ∣ cos (4 x) ∣ + 8 x^{2})

= 64 (\frac{1}{2} x^{2} sec^{2} (4 x) - \frac{1}{16} (8 x^{2} tan^{2} (4 x) - 4 x tan (4 x) - ln ∣ cos (4 x) ∣ + 8 x^{2}))

Füge eine Konstante zur Lösung hinzu

= 64 (\frac{1}{2} x^{2} sec^{2} (4 x) - \frac{1}{16} (8 x^{2} tan^{2} (4 x) - 4 x tan (4 x) - ln ∣ cos (4 x) ∣ + 8 x^{2})) + C

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