integral of sec^2(xta)n^2x

Lösung

\int sec^{2} (x t a) n^{2} x d x

n^{2} \frac{1}{a ^{2} t ^{2}} (a t x tan (a t x) + ln ∣ cos (a t x) ∣) + C

Schritte zur Lösung

\int sec^{2} (x t a) n^{2} x d x

Entferne die Konstante:

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

= n^{2} \cdot \int sec^{2} (a t x) x d x

Wende U-Substitution an

= n^{2} \cdot \int \frac{u sec ^{2} ( u )}{a ^{2} t ^{2}} d u

Entferne die Konstante:

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

= n^{2} \frac{1}{a ^{2} t ^{2}} \cdot \int u sec^{2} (u) d u

Wende die partielle Integration an

= n^{2} \frac{1}{a ^{2} t ^{2}} (u tan (u) - \int tan (u) d u)

\int tan (u) d u = - ln ∣ cos (u) ∣

= n^{2} \frac{1}{a ^{2} t ^{2}} (u tan (u) - (- ln ∣ cos (u) ∣))

Setze in

u = a t x

ein

= n^{2} \frac{1}{a ^{2} t ^{2}} (a t x tan (a t x) - (- ln ∣ cos (a t x) ∣))

Vereinfache

= n^{2} \frac{1}{a ^{2} t ^{2}} (a t x tan (a t x) + ln ∣ cos (a t x) ∣)

Füge eine Konstante zur Lösung hinzu

= n^{2} \frac{1}{a ^{2} t ^{2}} (a t x tan (a t x) + ln ∣ cos (a t x) ∣) + C

t an g e n t f (x) = x^{3} - \frac{3}{2} x^{2} + x, at x = 1

n = 2 \sum \infty \frac{( - 1 ) ^{n}}{n ln ( n )}

\int 4 + y^{2} d y

\int \frac{14}{( x - 1 ) ^{2} ( x - 2 ) ^{2}} d x

f (x) = arcsin (4 x)