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

Lösung

\int_{- 3}^{3} x^{2} + y^{2} d y

3 x^{2} + 9 + \frac{x ^{2} ( ln ( \frac{∣ 3 + x ^{2} + 9 ∣}{∣ x ∣} ) - ln ( \frac{∣ - 3 + x ^{2} + 9 ∣}{∣ x ∣} ) )}{2}

Schritte zur Lösung

\int_{- 3}^{3} x^{2} + y^{2} d y

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Wende integrale Reduktion an

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

\int_{a r c t a n (- \frac{3}{x})}^{a r c t a n (\frac{3}{x})} sec (u) d u = ln (\frac{3 + x ^{2} + 9}{∣ x ∣}) - ln (\frac{- 3 + x ^{2} + 9}{∣ x ∣})

= x^{2} [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (- \frac{3}{x})}^{a r c t a n (\frac{3}{x})} + \frac{1}{2} (ln (\frac{3 + x ^{2} + 9}{∣ x ∣}) - ln (\frac{- 3 + x ^{2} + 9}{∣ x ∣}))

Vereinfache

x^{2} [\frac{sec ^{2} ( u ) sin ( u )}{2}]_{a r c t a n (- \frac{3}{x})}^{a r c t a n (\frac{3}{x})} + \frac{1}{2} (ln (\frac{3 + x ^{2} + 9}{∣ x ∣}) - ln (\frac{- 3 + x ^{2} + 9}{∣ x ∣})) : x^{2} ([\frac{1}{2} sec (u) tan (u)]_{a r c t a n (- \frac{3}{x})}^{a r c t a n (\frac{3}{x})} + \frac{1}{2} (ln (\frac{3 + x ^{2} + 9}{∣ x ∣}) - ln (\frac{- 3 + x ^{2} + 9}{∣ x ∣})))

= x^{2} ([\frac{1}{2} sec (u) tan (u)]_{a r c t a n (- \frac{3}{x})}^{a r c t a n (\frac{3}{x})} + \frac{1}{2} (ln (\frac{3 + x ^{2} + 9}{∣ x ∣}) - ln (\frac{- 3 + x ^{2} + 9}{∣ x ∣})))

Berechne die Grenzen:

\frac{3 x ^{2} + 9}{x ^{2}}

= x^{2} (\frac{3 x ^{2} + 9}{x ^{2}} + \frac{1}{2} (ln (\frac{3 + x ^{2} + 9}{∣ x ∣}) - ln (\frac{- 3 + x ^{2} + 9}{∣ x ∣})))

Vereinfache

= 3 x^{2} + 9 + \frac{x ^{2} ( ln ( \frac{∣ 3 + x ^{2} + 9 ∣}{∣ x ∣} ) - ln ( \frac{∣ - 3 + x ^{2} + 9 ∣}{∣ x ∣} ) )}{2}

x \to 4 lim (3 x - 7)

d er i v a t i v e sin (x) + csc (x)

\int \frac{x}{3 x + 6} d x

l a pl a ce t r an s f or m sin (2 t) \cdot cos (2 t)

y^{^{''}} + y^{^{'}} = 2