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 2 of x/(x^2+6x+8)

解

\int_{0}^{2} \frac{x}{x ^{2} + 6 x + 8} d x

解

\frac{ln ( 3 ) + 3 ln ( \frac{3}{4} )}{2}

+1

十進法表記

0.11778 \dots

解答ステップ

\int_{0}^{2} \frac{x}{x ^{2} + 6 x + 8} d x

平方完成する

x^{2} + 6 x + 8 : (x + 3)^{2} - 1

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

u置換積分法を適用する

= \int_{3}^{5} \frac{u - 3}{u ^{2} - 1} d u

拡張

\frac{u - 3}{u ^{2} - 1} : \frac{u}{u ^{2} - 1} - \frac{3}{u ^{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

= \int_{3}^{5} \frac{u}{u ^{2} - 1} d u - \int_{3}^{5} \frac{3}{u ^{2} - 1} d u

\int_{3}^{5} \frac{u}{u ^{2} - 1} d u = \frac{1}{2} ln (3)

\int_{3}^{5} \frac{3}{u ^{2} - 1} d u = - \frac{3 ln ( \frac{3}{4} )}{2}

= \frac{1}{2} ln (3) - (- \frac{3 ln ( \frac{3}{4} )}{2})

簡素化

\frac{1}{2} ln (3) - (- \frac{3 ln ( \frac{3}{4} )}{2}) : \frac{ln ( 3 ) + 3 ln ( \frac{3}{4} )}{2}

= \frac{ln ( 3 ) + 3 ln ( \frac{3}{4} )}{2}

グラフ

人気の例

integral from-4 to 4 of [(32-x^2)-(x^2)]

\int_{- 4}^{4} [(32 - x^{2}) - (x^{2})] d x

integral from 0 to 0.41 of ln(1+x^2)

\int_{0}^{0.41} ln (1 + x^{2}) d x

integral from 4 to 5 of 1/(sqrt(5-x))

\int_{4}^{5} \frac{1}{5 - x} d x

integral from pi to 2pi of (1+cos(x))/x

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

integral from 1/10 to 11/10 of 1/x

\int_{\frac{1}{10}}^{\frac{11}{10}} \frac{1}{x} d x