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Beliebt Rechnen >

integral of (a^{arccot(x)}+ax) 1/(1+x^2)

Lösung

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

Lösung

a ln 1 + x^{2} - \frac{a ^{arccot (x)}}{ln ( a )} + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int - a cot (u) - a^{u} 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 a cot (u) d u - \int a^{u} d u

\int a cot (u) d u = a ln ∣ sin (u) ∣

\int a^{u} d u = \frac{a ^{u}}{ln ( a )}

= - a ln ∣ sin (u) ∣ - \frac{a ^{u}}{ln ( a )}

Setze in

u = arccot (x)

ein

= - a ln ∣ sin (arccot (x)) ∣ - \frac{a ^{arccot (x)}}{ln ( a )}

Vereinfache

- a ln ∣ sin (arccot (x)) ∣ - \frac{a ^{arccot (x)}}{ln ( a )} : a ln 1 + x^{2} - \frac{a ^{arccot (x)}}{ln ( a )}

= a ln 1 + x^{2} - \frac{a ^{arccot (x)}}{ln ( a )}

Füge eine Konstante zur Lösung hinzu

= a ln 1 + x^{2} - \frac{a ^{arccot (x)}}{ln ( a )} + C

Graph

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