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(10+16x^2)

Lösung

\int 10 + 16 x^{2} d x

Lösung

\frac{1}{4} (22 x 8 x^{2} + 5 + 5 (ln 22 x + 5 \frac{8 x ^{2} + 5}{5} - \frac{ln ( 5 )}{2})) + C

Schritte zur Lösung

\int 10 + 16 x^{2} d x

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{4}{10} x)

ein

= \frac{5}{2} \frac{sec ^{2} ( arctan ( \frac{4}{10} x ) ) sin ( arctan ( \frac{4}{10} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{4}{10} x)) + sec (arctan (\frac{4}{10} x))

Vereinfache

\frac{5}{2} \frac{sec ^{2} ( arctan ( \frac{4}{10} x ) ) sin ( arctan ( \frac{4}{10} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{4}{10} x)) + sec (arctan (\frac{4}{10} x)) : \frac{1}{4} (22 x 8 x^{2} + 5 + 5 (ln 22 x + 5 \frac{8 x ^{2} + 5}{5} - \frac{ln ( 5 )}{2}))

= \frac{1}{4} (22 x 8 x^{2} + 5 + 5 (ln 22 x + 5 \frac{8 x ^{2} + 5}{5} - \frac{ln ( 5 )}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (22 x 8 x^{2} + 5 + 5 (ln 22 x + 5 \frac{8 x ^{2} + 5}{5} - \frac{ln ( 5 )}{2})) + C

Graph

Beliebte Beispiele

integral of e^{-x^2+1}

\int e^{- x^{2} + 1} d x

integral of sqrt(9-2x^2)

\int 9 - 2 x^{2} d x

integral of (x^{-1}+e^{3x}+2x)

\int (x^{- 1} + e^{3 x} + 2 x) d x

integral of (x+x^2)/(1+x^2)

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

integral of 1/(x*sqrt(1+x^2))

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