integral from-1 to 0 of (2t)/((2+t^2)^3)

Lösung

\int_{- 1}^{0} \frac{2 t}{( 2 + t ^{2} ) ^{3}} d t

- \frac{5}{72}

Dezimale

- 0.06944 \dots

Schritte zur Lösung

\int_{- 1}^{0} \frac{2 t}{( 2 + t ^{2} ) ^{3}} d t

Entferne die Konstante:

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

= 2 \cdot \int_{- 1}^{0} \frac{t}{( 2 + t ^{2} ) ^{3}} d t

Wende U-Substitution an

= 2 \cdot \int_{3}^{2} \frac{1}{2 u ^{3}} d u

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

= 2 (- \int_{2}^{3} \frac{1}{2 u ^{3}} d u)

Entferne die Konstante:

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

= 2 (- \frac{1}{2} \cdot \int_{2}^{3} \frac{1}{u ^{3}} d u)

Wende die Potenzregel an

= 2 (- \frac{1}{2} [- \frac{1}{2 u ^{2}}]_{2}^{3})

Vereinfache

2 (- \frac{1}{2} [- \frac{1}{2 u ^{2}}]_{2}^{3}) : - [- \frac{1}{2 u ^{2}}]_{2}^{3}

= - [- \frac{1}{2 u ^{2}}]_{2}^{3}

Berechne die Grenzen:

\frac{5}{72}

= - \frac{5}{72}

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