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integral of (sqrt(x^2+16))/(2x)

Lösung

\int \frac{x ^{2} + 16}{2 x} d x

- ln \frac{1}{4} x^{2} + 16 + 1 + ln \frac{1}{4} x^{2} + 16 - 1 + \frac{1}{2} x^{2} + 16 + C

Schritte zur Lösung

\int \frac{x ^{2} + 16}{2 x} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{x ^{2} + 16}{x} d x

Wende U-Substitution an

= \frac{1}{2} \cdot \int \frac{u ^{2}}{u ^{2} - 16} d u

Faktorisiere

u^{2} - 16 : - (- u^{2} + 16)

= \frac{1}{2} \cdot \int \frac{u ^{2}}{- ( - u ^{2} + 16 )} d u

Entferne die Konstante:

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

= \frac{1}{2} (- \int \frac{u ^{2}}{- u ^{2} + 16} d u)

Wende integrale Substitution an

= \frac{1}{2} (- \int \frac{4 v ^{2}}{- v ^{2} + 1} d v)

Entferne die Konstante:

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

= \frac{1}{2} (- 4 \cdot \int \frac{v ^{2}}{- v ^{2} + 1} d v)

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= \frac{1}{2} (- 4 \cdot \int \frac{1}{- v ^{2} + 1} - 1 d v)

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (- 4 (\int \frac{1}{- v ^{2} + 1} d v - \int 1 d v))

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

\int 1 d v = v

= \frac{1}{2} (- 4 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v))

Ersetze zurück

= \frac{1}{2} - 4 \frac{ln \frac{x ^{2} + 16}{4} + 1}{2} - \frac{ln \frac{x ^{2} + 16}{4} - 1}{2} - \frac{x ^{2} + 16}{4}

Vereinfache

\frac{1}{2} - 4 \frac{ln \frac{x ^{2} + 16}{4} + 1}{2} - \frac{ln \frac{x ^{2} + 16}{4} - 1}{2} - \frac{x ^{2} + 16}{4} : - ln \frac{1}{4} x^{2} + 16 + 1 + ln \frac{1}{4} x^{2} + 16 - 1 + \frac{1}{2} x^{2} + 16

= - ln \frac{1}{4} x^{2} + 16 + 1 + ln \frac{1}{4} x^{2} + 16 - 1 + \frac{1}{2} x^{2} + 16

Füge eine Konstante zur Lösung hinzu

= - ln \frac{1}{4} x^{2} + 16 + 1 + ln \frac{1}{4} x^{2} + 16 - 1 + \frac{1}{2} x^{2} + 16 + C

\frac{d}{d x} (e^{3 x + 9})

d er i v a t i v e ln (x + 3)

d er i v a t i v e 10 + \frac{50 t}{2 t ^{2} + 9}

h \to 0 lim ((x + h)^{3} - x^{3})

\frac{\partial}{\partial t} (2 e^{s i n (x + 2 c t)})