integral of 8tan^4(xse)c^6x

Lösung

\int 8 tan^{4} (x se) c^{6} x d x

\frac{4 c ^{6}}{3 e ^{2} s ^{2}} (3 e^{2} s^{2} x^{2} + 2 es x tan^{3} (es x) - 6 es x tan (es x) + 2 ln ∣ sec (es x) ∣ - sec^{2} (es x) - 6 ln ∣ cos (es x) ∣) + C

Schritte zur Lösung

\int 8 tan^{4} (x se) c^{6} x d x

Entferne die Konstante:

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

= 8 c^{6} \cdot \int tan^{4} (se x) x d x

Wende U-Substitution an

= 8 c^{6} \cdot \int \frac{u tan ^{4} ( u )}{e ^{2} s ^{2}} d u

Entferne die Konstante:

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

= 8 c^{6} \frac{1}{e ^{2} s ^{2}} \cdot \int u tan^{4} (u) d u

Wende die partielle Integration an

= 8 c^{6} \frac{1}{e ^{2} s ^{2}} (u (\frac{1}{3} tan^{3} (u) + u - tan (u)) - \int \frac{1}{3} tan^{3} (u) + u - tan (u) d u)

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

= 8 c^{6} \frac{1}{e ^{2} s ^{2}} (u (\frac{1}{3} tan^{3} (u) + u - tan (u)) - (\frac{1}{3} (- ln ∣ sec (u) ∣ + \frac{1}{2} sec^{2} (u)) + \frac{u ^{2}}{2} + ln ∣ cos (u) ∣))

Setze in

u = se x

ein

= 8 c^{6} \frac{1}{e ^{2} s ^{2}} (se x (\frac{1}{3} tan^{3} (se x) + se x - tan (se x)) - (\frac{1}{3} (- ln ∣ sec (se x) ∣ + \frac{1}{2} sec^{2} (se x)) + \frac{( se x ) ^{2}}{2} + ln ∣ cos (se x) ∣))

Vereinfache

8 c^{6} \frac{1}{e ^{2} s ^{2}} (se x (\frac{1}{3} tan^{3} (se x) + se x - tan (se x)) - (\frac{1}{3} (- ln ∣ sec (se x) ∣ + \frac{1}{2} sec^{2} (se x)) + \frac{( se x ) ^{2}}{2} + ln ∣ cos (se x) ∣)) : \frac{4 c ^{6}}{3 e ^{2} s ^{2}} (3 e^{2} s^{2} x^{2} + 2 es x tan^{3} (es x) - 6 es x tan (es x) + 2 ln ∣ sec (es x) ∣ - sec^{2} (es x) - 6 ln ∣ cos (es x) ∣)

= \frac{4 c ^{6}}{3 e ^{2} s ^{2}} (3 e^{2} s^{2} x^{2} + 2 es x tan^{3} (es x) - 6 es x tan (es x) + 2 ln ∣ sec (es x) ∣ - sec^{2} (es x) - 6 ln ∣ cos (es x) ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{4 c ^{6}}{3 e ^{2} s ^{2}} (3 e^{2} s^{2} x^{2} + 2 es x tan^{3} (es x) - 6 es x tan (es x) + 2 ln ∣ sec (es x) ∣ - sec^{2} (es x) - 6 ln ∣ cos (es x) ∣) + C

d er i v a t i v e e^{\frac{1}{2} x}

a re a 4 - 4 cos (7 x), 4 cos (7 x), 0 \geq x \geq \frac{π}{7}

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

\int x^{2} + 7 d x

y^{^{''}} = - 3 y - 2 y^{^{'}}, y (0) = 1, y^{^{'}} (0) = 0