integral of (a+x)/(sqrt(a^2-x^2))

Lösung

\int \frac{a + x}{a ^{2} - x ^{2}} d x

a arcsin (\frac{x}{a}) - a^{2} - x^{2} + C

Schritte zur Lösung

\int \frac{a + x}{a ^{2} - x ^{2}} d x

Multipliziere aus

\frac{a + x}{a ^{2} - x ^{2}} : \frac{a}{a ^{2} - x ^{2}} + \frac{x}{a ^{2} - x ^{2}}

= \int \frac{a}{a ^{2} - x ^{2}} + \frac{x}{a ^{2} - x ^{2}} d x

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{a}{a ^{2} - x ^{2}} d x + \int \frac{x}{a ^{2} - x ^{2}} d x

\int \frac{a}{a ^{2} - x ^{2}} d x = a arcsin (\frac{x}{a})

\int \frac{x}{a ^{2} - x ^{2}} d x = - a^{2} - x^{2}

= a arcsin (\frac{x}{a}) - a^{2} - x^{2}

Füge eine Konstante zur Lösung hinzu

= a arcsin (\frac{x}{a}) - a^{2} - x^{2} + C

\int x (1 - x)^{5} d x

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\int \frac{1}{10} x^{3} d x

\int \frac{1}{2 x ^{\frac{1}{3}}} d x