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Beliebt Rechnen >

integral of (tan(x)+1/(tan(x)))^2

Lösung

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

Lösung

- 2 cot (2 x) + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

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

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

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

\int 2 d x = 2 x

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

= tan (x) - x + 2 x - cot (x) - x

Vereinfache

tan (x) - x + 2 x - cot (x) - x : - 2 cot (2 x)

= - 2 cot (2 x)

Füge eine Konstante zur Lösung hinzu

= - 2 cot (2 x) + C

Graph

Beliebte Beispiele

x^2y^{''}-2y=sin(ln(x))

x^{2} y^{^{''}} - 2 y = sin (ln (x))

limit as x approaches-2 of 2x

x \to - 2 lim (2 x)

integral of 1/(5x^7)

\int \frac{1}{5 x ^{7}} d x

maclaurin (arctan(x))/(x+1)

ma c l a u r in \frac{arctan ( x )}{x + 1}

(\partial)/(\partial y)(y-cos(y))

\frac{\partial}{\partial y} (y - cos (y))