integral of 1/(cos^4(x)sin^2(x))

Lösung

\int \frac{1}{cos ^{4} ( x ) sin ^{2} ( x )} d x

- cot (x) + 2 tan (x) + \frac{1}{3} tan^{3} (x) + C

Schritte zur Lösung

\int \frac{1}{cos ^{4} ( x ) sin ^{2} ( x )} d x

Trigonometrische Substitution anwenden

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

Multipliziere aus

\frac{( 1 + u ^{2} ) ^{2}}{u ^{2}} : \frac{1}{u ^{2}} + 2 + u^{2}

= \int \frac{1}{u ^{2}} + 2 + u^{2} 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

= \int \frac{1}{u ^{2}} d u + \int 2 d u + \int u^{2} d u

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

\int 2 d u = 2 u

\int u^{2} d u = \frac{u ^{3}}{3}

= - \frac{1}{u} + 2 u + \frac{u ^{3}}{3}

Setze in

u = tan (x)

ein

= - \frac{1}{tan ( x )} + 2 tan (x) + \frac{tan ^{3} ( x )}{3}

Vereinfache

- \frac{1}{tan ( x )} + 2 tan (x) + \frac{tan ^{3} ( x )}{3} : - cot (x) + 2 tan (x) + \frac{1}{3} tan^{3} (x)

= - cot (x) + 2 tan (x) + \frac{1}{3} tan^{3} (x)

Füge eine Konstante zur Lösung hinzu

= - cot (x) + 2 tan (x) + \frac{1}{3} tan^{3} (x) + C

\int (10 x^{4} - 4 x^{2} - 14 x + 12) d x

\int b cos (a t) d t

\int \frac{1}{500} e^{- \frac{x}{500}} d x

\int \frac{3}{4 x ^{2} + 16} d x

\int \frac{x ^{2} + x + 1}{x ^{2}} d x