integral of (1+tan(x))^2*sec(x)

Lösung

\int (1 + tan (x))^{2} \cdot sec (x) d x

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

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

\frac{( cos ( x ) + sin ( x ) ) ^{2}}{cos ^{3} ( x )} : \frac{1}{cos ( x )} + \frac{2 sin ( x )}{cos ^{2} ( x )} + \frac{sin ^{2} ( x )}{cos ^{3} ( x )}

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

\int \frac{1}{cos ( x )} d x = ln ∣ tan (x) + sec (x) ∣

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

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

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

Vereinfache

ln ∣ tan (x) + sec (x) ∣ + 2 sec (x) - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ : - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + 2 sec (x)

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

Füge eine Konstante zur Lösung hinzu

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

\int (tan (x))^{3} (sec (x))^{6} d x

\int 5 - y d y

\int x^{5} x^{3} + 8 d x

\int 7 e^{x} + 6 sec^{2} (x) d x

\int \frac{1}{( 2 x + 7 ) ^{3}} d x