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Beliebt Rechnen >

integral of (msec(θ)-tan(θ))^2

Lösung

\int (m sec (θ) - tan (θ))^{2} d θ

Lösung

tan (θ) - θ - 2 m sec (θ) + m^{2} tan (θ) + C

Schritte zur Lösung

\int (m sec (θ) - tan (θ))^{2} d θ

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{( - sin ( θ ) + m ) ^{2}}{cos ^{2} ( θ )} d θ

Multipliziere aus

\frac{( - sin ( θ ) + m ) ^{2}}{cos ^{2} ( θ )} : \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )} - \frac{2 m sin ( θ )}{cos ^{2} ( θ )} + \frac{m ^{2}}{cos ^{2} ( θ )}

= \int \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )} - \frac{2 m sin ( θ )}{cos ^{2} ( θ )} + \frac{m ^{2}}{cos ^{2} ( θ )} d θ

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{sin ^{2} ( θ )}{cos ^{2} ( θ )} d θ - \int \frac{2 m sin ( θ )}{cos ^{2} ( θ )} d θ + \int \frac{m ^{2}}{cos ^{2} ( θ )} d θ

\int \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )} d θ = tan (θ) - θ

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

\int \frac{m ^{2}}{cos ^{2} ( θ )} d θ = m^{2} tan (θ)

= tan (θ) - θ - 2 m sec (θ) + m^{2} tan (θ)

Füge eine Konstante zur Lösung hinzu

= tan (θ) - θ - 2 m sec (θ) + m^{2} tan (θ) + C

Graph

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