integral from 0 to pi/4 of xtan(x^2)

Lösung

\int_{0}^{\frac{π}{4}} x tan (x^{2}) d x

- \frac{1}{2} ln (cos (\frac{π ^{2}}{16}))

Dezimale

0.10185 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{4}} x tan (x^{2}) d x

Wende U-Substitution an

= \int_{0}^{\frac{π ^{2}}{16}} \frac{tan ( u )}{2} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{0}^{\frac{π ^{2}}{16}} tan (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{1}{2} \cdot \int_{0}^{\frac{π ^{2}}{16}} \frac{sin ( u )}{cos ( u )} d u

Wende U-Substitution an

= \frac{1}{2} \cdot \int_{1}^{c o s (\frac{π ^{2}}{16})} - \frac{1}{v} d v

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= \frac{1}{2} (- \int_{c o s (\frac{π ^{2}}{16})}^{1} - \frac{1}{v} d v)

Entferne die Konstante:

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

= \frac{1}{2} (- (- \int_{c o s (\frac{π ^{2}}{16})}^{1} \frac{1}{v} d v))

Nutze das gemeinsame Integral :

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} (- (- [ln ∣ v ∣]_{c o s (\frac{π ^{2}}{16})}^{1}))

Vereinfache

= \frac{1}{2} [ln ∣ v ∣]_{c o s (\frac{π ^{2}}{16})}^{1}

Berechne die Grenzen:

- ln (cos (\frac{π ^{2}}{16}))

= \frac{1}{2} (- ln (cos (\frac{π ^{2}}{16})))

Vereinfache

= - \frac{1}{2} ln (cos (\frac{π ^{2}}{16}))

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