integral from 7 to 8 of x/(x^2+4x+20)

Lösung

\int_{7}^{8} \frac{x}{x ^{2} + 4 x + 20} d x

\frac{1}{2} (ln (\frac{116}{97}) - arctan (\frac{5}{2}) + arctan (\frac{9}{4}))

Dezimale

0.07058 \dots

Schritte zur Lösung

\int_{7}^{8} \frac{x}{x ^{2} + 4 x + 20} d x

Vervollständige das Quadrat

x^{2} + 4 x + 20 : (x + 2)^{2} + 16

= \int_{7}^{8} \frac{x}{( x + 2 ) ^{2} + 16} d x

Wende U-Substitution an

= \int_{9}^{10} \frac{u - 2}{u ^{2} + 16} d u

Wende integrale Substitution an

= \int_{\frac{9}{4}}^{\frac{5}{2}} \frac{2 v - 1}{2 ( v ^{2} + 1 )} d v

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{\frac{9}{4}}^{\frac{5}{2}} \frac{2 v - 1}{v ^{2} + 1} d v

Multipliziere aus

\frac{2 v - 1}{v ^{2} + 1} : \frac{2 v}{v ^{2} + 1} - \frac{1}{v ^{2} + 1}

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}{2} (\int_{\frac{9}{4}}^{\frac{5}{2}} \frac{2 v}{v ^{2} + 1} d v - \int_{\frac{9}{4}}^{\frac{5}{2}} \frac{1}{v ^{2} + 1} d v)

\int_{\frac{9}{4}}^{\frac{5}{2}} \frac{2 v}{v ^{2} + 1} d v = ln (\frac{116}{97})

\int_{\frac{9}{4}}^{\frac{5}{2}} \frac{1}{v ^{2} + 1} d v = arctan (\frac{5}{2}) - arctan (\frac{9}{4})

= \frac{1}{2} (ln (\frac{116}{97}) - (arctan (\frac{5}{2}) - arctan (\frac{9}{4})))

Vereinfache

= \frac{1}{2} (ln (\frac{116}{97}) - arctan (\frac{5}{2}) + arctan (\frac{9}{4}))

\int y z^{2} d y

\int (4 + e^{x}) d x

\int \frac{e ^{3 x}}{9 + e ^{2 x}} d x

\int_{0}^{3} (\frac{6}{1 + x ^{2}}) d x

x \to 0 lim ((3 x - 4)^{4})