integral of (tan(4x)+sec(4x))^2

Lösung

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

\frac{1}{2} (- 2 x + tan (4 x) + sec (4 x)) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int (tan (u) + sec (u))^{2} \frac{1}{4} d u

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int (tan (u) + sec (u))^{2} d u

Multipliziere aus

(tan (u) + sec (u))^{2} : tan^{2} (u) + 2 tan (u) sec (u) + sec^{2} (u)

= \frac{1}{4} \cdot \int tan^{2} (u) + 2 tan (u) sec (u) + sec^{2} (u) d u

Wende die Summenregel an:

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

= \frac{1}{4} (\int tan^{2} (u) d u + \int 2 tan (u) sec (u) d u + \int sec^{2} (u) d u)

\int tan^{2} (u) d u = - u + tan (u)

\int 2 tan (u) sec (u) d u = 2 sec (u)

\int sec^{2} (u) d u = tan (u)

= \frac{1}{4} (- u + tan (u) + 2 sec (u) + tan (u))

Setze in

u = 4 x

ein

= \frac{1}{4} (- 4 x + tan (4 x) + 2 sec (4 x) + tan (4 x))

Vereinfache

\frac{1}{4} (- 4 x + tan (4 x) + 2 sec (4 x) + tan (4 x)) : \frac{1}{2} (- 2 x + tan (4 x) + sec (4 x))

= \frac{1}{2} (- 2 x + tan (4 x) + sec (4 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (- 2 x + tan (4 x) + sec (4 x)) + C

\frac{\partial}{\partial x} (e^{3 x - y})

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n = 0 \sum \infty - \frac{1}{5 ^{n}}

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x \to - 5 lim (\frac{x + 5}{x + 5})