ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral from 0 to 1 of (2x)/(x^2+4x+29)

解

\int_{0}^{1} \frac{2 x}{x ^{2} + 4 x + 29} d x

解

\frac{5 ( ln ( \frac{34}{25} ) - ln ( \frac{29}{25} ) ) - 4 ( arctan ( \frac{3}{5} ) - arctan ( \frac{2}{5} ) )}{5}

+1

十進法表記

0.03113 \dots

解答ステップ

\int_{0}^{1} \frac{2 x}{x ^{2} + 4 x + 29} d x

定数を除く:

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

= 2 \cdot \int_{0}^{1} \frac{x}{x ^{2} + 4 x + 29} d x

平方完成する

x^{2} + 4 x + 29 : (x + 2)^{2} + 25

= 2 \cdot \int_{0}^{1} \frac{x}{( x + 2 ) ^{2} + 25} d x

u置換積分法を適用する

= 2 \cdot \int_{2}^{3} \frac{u - 2}{u ^{2} + 25} d u

置換積分法を適用する

= 2 \cdot \int_{\frac{2}{5}}^{\frac{3}{5}} \frac{5 v - 2}{5 ( v ^{2} + 1 )} d v

定数を除く:

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

= 2 \cdot \frac{1}{5} \cdot \int_{\frac{2}{5}}^{\frac{3}{5}} \frac{5 v - 2}{v ^{2} + 1} d v

拡張

\frac{5 v - 2}{v ^{2} + 1} : \frac{5 v}{v ^{2} + 1} - \frac{2}{v ^{2} + 1}

総和規則を適用する:

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

= 2 \cdot \frac{1}{5} (\int_{\frac{2}{5}}^{\frac{3}{5}} \frac{5 v}{v ^{2} + 1} d v - \int_{\frac{2}{5}}^{\frac{3}{5}} \frac{2}{v ^{2} + 1} d v)

\int_{\frac{2}{5}}^{\frac{3}{5}} \frac{5 v}{v ^{2} + 1} d v = \frac{5}{2} (ln (\frac{34}{25}) - ln (\frac{29}{25}))

\int_{\frac{2}{5}}^{\frac{3}{5}} \frac{2}{v ^{2} + 1} d v = 2 (arctan (\frac{3}{5}) - arctan (\frac{2}{5}))

= 2 \cdot \frac{1}{5} (\frac{5}{2} (ln (\frac{34}{25}) - ln (\frac{29}{25})) - 2 (arctan (\frac{3}{5}) - arctan (\frac{2}{5})))

簡素化

2 \cdot \frac{1}{5} (\frac{5}{2} (ln (\frac{34}{25}) - ln (\frac{29}{25})) - 2 (arctan (\frac{3}{5}) - arctan (\frac{2}{5}))) : \frac{5 ( ln ( \frac{34}{25} ) - ln ( \frac{29}{25} ) ) - 4 ( arctan ( \frac{3}{5} ) - arctan ( \frac{2}{5} ) )}{5}

= \frac{5 ( ln ( \frac{34}{25} ) - ln ( \frac{29}{25} ) ) - 4 ( arctan ( \frac{3}{5} ) - arctan ( \frac{2}{5} ) )}{5}

グラフ

人気の例

integral from-infinity to 0 of e^{17x}

\int_{- \infty}^{0} e^{17 x} d x

integral from 1 to 2 of x^2(3/7 x^2)

\int_{1}^{2} x^{2} (\frac{3}{7} x^{2}) d x

integral from 0 to pi/2 of x^3*cos(2x)

\int_{0}^{\frac{π}{2}} x^{3} \cdot cos (2 x) d x

integral from 1 to 4 of 3x^2+xsqrt(x)+1

\int_{1}^{4} 3 x^{2} + x x + 1 d x

integral from-1 to 1 of 1/4

\int_{- 1}^{1} \frac{1}{4} d x