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Beliebt Rechnen >

integral from 1 to 5 of sqrt(1+x^2)

Lösung

\int_{1}^{5} 1 + x^{2} d x

Lösung

\frac{2 ( - 1 + 5 13 ) - ln ( 1 + 2 ) + ln ( 5 + 26 )}{2}

+1

Dezimale

12.75597 \dots

Schritte zur Lösung

\int_{1}^{5} 1 + x^{2} d x

Trigonometrische Substitution anwenden

= \int_{\frac{π}{4}}^{a r c t a n (5)} sec^{3} (u) d u

Wende integrale Reduktion an

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{4}}^{a r c t a n (5)} + \frac{1}{2} \cdot \int_{\frac{π}{4}}^{a r c t a n (5)} sec (u) d u

\int_{\frac{π}{4}}^{a r c t a n (5)} sec (u) d u = ln (5 + 26) - ln (1 + 2)

= [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{4}}^{a r c t a n (5)} + \frac{1}{2} (ln (5 + 26) - ln (1 + 2))

Vereinfache

[\frac{sec ^{2} ( u ) sin ( u )}{2}]_{\frac{π}{4}}^{a r c t a n (5)} + \frac{1}{2} (ln (5 + 26) - ln (1 + 2)) : [\frac{1}{2} sec (u) tan (u)]_{\frac{π}{4}}^{a r c t a n (5)} + \frac{1}{2} (ln (5 + 26) - ln (1 + 2))

= [\frac{1}{2} sec (u) tan (u)]_{\frac{π}{4}}^{a r c t a n (5)} + \frac{1}{2} (ln (5 + 26) - ln (1 + 2))

Berechne die Grenzen:

\frac{5 13 - 1}{2}

= \frac{5 13 - 1}{2} + \frac{1}{2} (ln (5 + 26) - ln (1 + 2))

Vereinfache

= \frac{2 ( - 1 + 5 13 ) - ln ( 1 + 2 ) + ln ( 5 + 26 )}{2}

Graph

Beliebte Beispiele

integral from 3 to 7 of 1/x

\int_{3}^{7} \frac{1}{x} d x

integral from 5 to x of t^3

\int_{5}^{x} t^{3} d t

integral from 0 to pi/2 of (1-sin(x))^2

\int_{0}^{\frac{π}{2}} (1 - sin (x))^{2} d x

integral from 0 to y of t

\int_{0}^{y} t d t

integral from 0 to pi/2 of 1/(cos(x)+2)

\int_{0}^{\frac{π}{2}} \frac{1}{cos ( x ) + 2} d x