integral from 1 to 2 of 7sqrt(4x^2-3)

Lösung

\int_{1}^{2} 7 4 x^{2} - 3 d x

\frac{7 ( 3 ln ( 3 ) + 4 13 - 3 ln ( 4 + 13 ) - 2 )}{4}

Dezimale

16.85496 \dots

Schritte zur Lösung

\int_{1}^{2} 7 4 x^{2} - 3 d x

Entferne die Konstante:

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

= 7 \cdot \int_{1}^{2} 4 x^{2} - 3 d x

Trigonometrische Substitution anwenden

= 7 \cdot \int_{\frac{π}{6}}^{arcsec (\frac{4}{3})} \frac{3}{2} tan^{2} (u) sec (u) d u

Entferne die Konstante:

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

= 7 \cdot \frac{3}{2} \cdot \int_{\frac{π}{6}}^{arcsec (\frac{4}{3})} tan^{2} (u) sec (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 7 \cdot \frac{3}{2} \cdot \int_{\frac{π}{6}}^{arcsec (\frac{4}{3})} (- 1 + sec^{2} (u)) sec (u) d u

Multipliziere aus

(- 1 + sec^{2} (u)) sec (u) : - sec (u) + sec^{3} (u)

= 7 \cdot \frac{3}{2} \cdot \int_{\frac{π}{6}}^{arcsec (\frac{4}{3})} - sec (u) + sec^{3} (u) 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

= 7 \cdot \frac{3}{2} (- \int_{\frac{π}{6}}^{arcsec (\frac{4}{3})} sec (u) d u + \int_{\frac{π}{6}}^{arcsec (\frac{4}{3})} sec^{3} (u) d u)

\int_{\frac{π}{6}}^{arcsec (\frac{4}{3})} sec (u) d u = ln (\frac{13}{3} + \frac{4}{3}) - \frac{1}{2} ln (3)

\int_{\frac{π}{6}}^{arcsec (\frac{4}{3})} sec^{3} (u) d u = \frac{2 13 - 1}{3} + \frac{1}{2} (ln (\frac{13}{3} + \frac{4}{3}) - \frac{1}{2} ln (3))

= 7 \cdot \frac{3}{2} (- (ln (\frac{13}{3} + \frac{4}{3}) - \frac{1}{2} ln (3)) + \frac{2 13 - 1}{3} + \frac{1}{2} (ln (\frac{13}{3} + \frac{4}{3}) - \frac{1}{2} ln (3)))

Vereinfache

7 \cdot \frac{3}{2} (- (ln (\frac{13}{3} + \frac{4}{3}) - \frac{1}{2} ln (3)) + \frac{2 13 - 1}{3} + \frac{1}{2} (ln (\frac{13}{3} + \frac{4}{3}) - \frac{1}{2} ln (3))) : \frac{7 ( 3 ln ( 3 ) + 4 13 - 3 ln ( 4 + 13 ) - 2 )}{4}

= \frac{7 ( 3 ln ( 3 ) + 4 13 - 3 ln ( 4 + 13 ) - 2 )}{4}

\int x^{- 10} d x

x \to 2 lim (\frac{- 1}{x - 2})

l a pl a ce t r an s f or m 3 t sin (3 t)

\frac{d ^{2} y}{d x ^{2}} + x \frac{d y}{d x} = 0

y^{^{'}} = (x - 4) e^{- 2 y}, y (4) = ln (4)