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Beliebt Rechnen >

integral from-1 to x^3 of sqrt(2+t^2)

Lösung

\int_{- 1}^{x^{3}} 2 + t^{2} d t

Lösung

\frac{x ^{3} x ^{6} + 2 + 3}{2} + ln \frac{x ^{3}}{2} + \frac{2 + x ^{6}}{2} - ln (- \frac{1}{2} + \frac{3}{2})

Schritte zur Lösung

\int_{- 1}^{x^{3}} 2 + t^{2} d t

Trigonometrische Substitution anwenden

= \int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{x ^{3}}{2})} 2 sec^{3} (u) d u

Entferne die Konstante:

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

= 2 \cdot \int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{x ^{3}}{2})} sec^{3} (u) d u

Wende integrale Reduktion an

= 2 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{x ^{3}}{2})} + \frac{1}{2} \cdot \int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{x ^{3}}{2})} sec (u) d u

\int_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{x ^{3}}{2})} sec (u) d u = ln \frac{x ^{3}}{2} + \frac{2 + x ^{6}}{2} - ln (- \frac{1}{2} + \frac{3}{2})

= 2 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{x ^{3}}{2})} + \frac{1}{2} (ln \frac{x ^{3}}{2} + \frac{2 + x ^{6}}{2} - ln (- \frac{1}{2} + \frac{3}{2}))

Vereinfache

2 [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{x ^{3}}{2})} + \frac{1}{2} (ln \frac{x ^{3}}{2} + \frac{2 + x ^{6}}{2} - ln (- \frac{1}{2} + \frac{3}{2})) : 2 [\frac{1}{2} sec (u) tan (u)]_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{x ^{3}}{2})} + ln \frac{x ^{3}}{2} + \frac{2 + x ^{6}}{2} - ln (- \frac{1}{2} + \frac{3}{2})

= 2 [\frac{1}{2} sec (u) tan (u)]_{- a r c t a n (\frac{1}{2})}^{a r c t a n (\frac{x ^{3}}{2})} + ln \frac{x ^{3}}{2} + \frac{2 + x ^{6}}{2} - ln (- \frac{1}{2} + \frac{3}{2})

Berechne die Grenzen:

\frac{x ^{3} 2 + x ^{6}}{4} + \frac{3}{4}

= 2 (\frac{x ^{3} 2 + x ^{6}}{4} + \frac{3}{4}) + ln \frac{x ^{3}}{2} + \frac{2 + x ^{6}}{2} - ln (- \frac{1}{2} + \frac{3}{2})

Vereinfache

= \frac{x ^{3} x ^{6} + 2 + 3}{2} + ln \frac{x ^{3}}{2} + \frac{2 + x ^{6}}{2} - ln (- \frac{1}{2} + \frac{3}{2})

Beliebte Beispiele

(x^2+1)y^'+6x(y-1)=0,y(0)=9

(x^{2} + 1) y^{^{'}} + 6 x (y - 1) = 0, y (0) = 9

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