de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of sqrt(4t^2+5)

Lösung

\int 4 t^{2} + 5 d t

Lösung

\frac{1}{4} (2 t 4 t^{2} + 5 + 5 (ln 2 t + 5 \frac{4 t ^{2} + 5}{5} - \frac{ln ( 5 )}{2})) + C

Schritte zur Lösung

\int 4 t^{2} + 5 d t

Trigonometrische Substitution anwenden

= \int \frac{5}{2} sec^{3} (u) d u

Entferne die Konstante:

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

= \frac{5}{2} \cdot \int sec^{3} (u) d u

Wende integrale Reduktion an

= \frac{5}{2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= \frac{5}{2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

u = arctan (\frac{2}{5} t)

ein

= \frac{5}{2} \frac{sec ^{2} ( arctan ( \frac{2}{5} t ) ) sin ( arctan ( \frac{2}{5} t ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{2}{5} t)) + sec (arctan (\frac{2}{5} t))

Vereinfache

\frac{5}{2} \frac{sec ^{2} ( arctan ( \frac{2}{5} t ) ) sin ( arctan ( \frac{2}{5} t ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{2}{5} t)) + sec (arctan (\frac{2}{5} t)) : \frac{1}{4} (2 t 4 t^{2} + 5 + 5 (ln 2 t + 5 \frac{4 t ^{2} + 5}{5} - \frac{ln ( 5 )}{2}))

= \frac{1}{4} (2 t 4 t^{2} + 5 + 5 (ln 2 t + 5 \frac{4 t ^{2} + 5}{5} - \frac{ln ( 5 )}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (2 t 4 t^{2} + 5 + 5 (ln 2 t + 5 \frac{4 t ^{2} + 5}{5} - \frac{ln ( 5 )}{2})) + C

Graph

Beliebte Beispiele

integral of (x^2-3)(x+2)

\int (x^{2} - 3) (x + 2) d x

integral of 3/(sqrt(x))+sin(2x)

\int \frac{3}{x} + sin (2 x) d x

integral of sqrt(x^2-8)

\int x^{2} - 8 d x

integral of x^{5/2}-x^{1/2}

\int x^{\frac{5}{2}} - x^{\frac{1}{2}} d x

integral of x^2(2x^3+4)

\int x^{2} (2 x^{3} + 4) d x