integral of x/(sqrt(x^2+2x+25))

Lösung

\int \frac{x}{x ^{2} + 2 x + 25} d x

x^{2} + 2 x + 25 - ln (\frac{1}{2 6} x + 1 + x^{2} + 2 x + 25) + C

Schritte zur Lösung

\int \frac{x}{x ^{2} + 2 x + 25} d x

Vervollständige das Quadrat

x^{2} + 2 x + 25 : (x + 1)^{2} + 24

= \int \frac{x}{( x + 1 ) ^{2} + 24} d x

Wende U-Substitution an

= \int \frac{u - 1}{u ^{2} + 24} d u

Multipliziere aus

\frac{u - 1}{u ^{2} + 24} : \frac{u}{u ^{2} + 24} - \frac{1}{u ^{2} + 24}

= \int \frac{u}{u ^{2} + 24} - \frac{1}{u ^{2} + 24} 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 \frac{u}{u ^{2} + 24} d u - \int \frac{1}{u ^{2} + 24} d u

\int \frac{u}{u ^{2} + 24} d u = u^{2} + 24

\int \frac{1}{u ^{2} + 24} d u = ln (\frac{1}{2 6} u + 24 + u^{2})

= u^{2} + 24 - ln (\frac{1}{2 6} u + 24 + u^{2})

Setze in

u = x + 1

ein

= (x + 1)^{2} + 24 - ln (\frac{1}{2 6} x + 1 + 24 + (x + 1)^{2})

Vereinfache

(x + 1)^{2} + 24 - ln (\frac{1}{2 6} x + 1 + 24 + (x + 1)^{2}) : x^{2} + 2 x + 25 - ln (\frac{1}{2 6} x + 1 + x^{2} + 2 x + 25)

= x^{2} + 2 x + 25 - ln (\frac{1}{2 6} x + 1 + x^{2} + 2 x + 25)

Füge eine Konstante zur Lösung hinzu

= x^{2} + 2 x + 25 - ln (\frac{1}{2 6} x + 1 + x^{2} + 2 x + 25) + C

\int \frac{5 x + 7}{4 x ^{2} + 12 x + 10} d x

d er i v a t i v e f (t) = \frac{( e ^{x} + 5 x )}{5 x}

a re a x^{2} + 2, 2 x + 6

\int 10 tan^{2} (x) d x

\int 14 x - 3 d x