integral of 20sqrt(tan(x))sec^4(x)

Lösung

\int 20 tan (x) sec^{4} (x) d x

20 (\frac{2}{3} tan^{\frac{3}{2}} (x) + \frac{2}{7} tan^{\frac{7}{2}} (x)) + C

Schritte zur Lösung

\int 20 tan (x) sec^{4} (x) d x

Entferne die Konstante:

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

= 20 \cdot \int tan (x) sec^{4} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 20 \cdot \int (1 + tan^{2} (x)) sec^{2} (x) tan (x) d x

Wende U-Substitution an

= 20 \cdot \int u (1 + u^{2}) d u

Multipliziere aus

u (1 + u^{2}) : u + u^{\frac{5}{2}}

= 20 \cdot \int u + u^{\frac{5}{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

= 20 (\int u d u + \int u^{\frac{5}{2}} d u)

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

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

= 20 (\frac{2}{3} u^{\frac{3}{2}} + \frac{2}{7} u^{\frac{7}{2}})

Setze in

u = tan (x)

ein

= 20 (\frac{2}{3} tan^{\frac{3}{2}} (x) + \frac{2}{7} tan^{\frac{7}{2}} (x))

Füge eine Konstante zur Lösung hinzu

= 20 (\frac{2}{3} tan^{\frac{3}{2}} (x) + \frac{2}{7} tan^{\frac{7}{2}} (x)) + C

\int \frac{6 x ^{8} + 9 x ^{2} - 9 x}{3 x ^{2}} d x

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

\int 4 x^{3} + sin (x) d x

\int \frac{3 x ^{3} - x ^{2} + 6 x + 24}{x ^{2} - 2 x + 5} d x

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