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

Lösung

\int_{0}^{3} \frac{1}{3 + 2 ^{x}} d x

\frac{5 ln ( 2 ) - ln ( 11 )}{3 ln ( 2 )}

Dezimale

0.51352 \dots

Schritte zur Lösung

\int_{0}^{3} \frac{1}{3 + 2 ^{x}} d x

Wende U-Substitution an

= \int_{1}^{8} \frac{1}{ln ( 2 ) u ( 3 + u )} d u

Entferne die Konstante:

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

= \frac{1}{ln ( 2 )} \cdot \int_{1}^{8} \frac{1}{u ( 3 + u )} d u

Ermittle den Partialbruch von

\frac{1}{u ( 3 + u )} : \frac{1}{3 u} - \frac{1}{3 ( u + 3 )}

= \frac{1}{ln ( 2 )} \cdot \int_{1}^{8} \frac{1}{3 u} - \frac{1}{3 ( u + 3 )} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{ln ( 2 )} (\int_{1}^{8} \frac{1}{3 u} d u - \int_{1}^{8} \frac{1}{3 ( u + 3 )} d u)

\int_{1}^{8} \frac{1}{3 u} d u = ln (2)

\int_{1}^{8} \frac{1}{3 ( u + 3 )} d u = \frac{1}{3} (ln (11) - 2 ln (2))

= \frac{1}{ln ( 2 )} (ln (2) - \frac{1}{3} (ln (11) - 2 ln (2)))

Vereinfache

\frac{1}{ln ( 2 )} (ln (2) - \frac{1}{3} (ln (11) - 2 ln (2))) : \frac{5 ln ( 2 ) - ln ( 11 )}{3 ln ( 2 )}

= \frac{5 ln ( 2 ) - ln ( 11 )}{3 ln ( 2 )}

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

\int_{1}^{3} \frac{1}{( x - 2 ) ^{\frac{5}{3}}} d x

\int_{0}^{π} 5 + sin (x) d x

\int_{3 π}^{4 π} cos (\frac{1}{2} x) d x

\int_{- 1}^{1} (5 x + 5) d x