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 3 to 8 of x/(x^2+6x+13)

解

\int_{3}^{8} \frac{x}{x ^{2} + 6 x + 13} d x

解

\frac{ln ( \frac{25}{8} ) - 3 ( arctan ( \frac{11}{2} ) - arctan ( 3 ) )}{2}

+1

十進法表記

0.35687 \dots

解答ステップ

\int_{3}^{8} \frac{x}{x ^{2} + 6 x + 13} d x

平方完成する

x^{2} + 6 x + 13 : (x + 3)^{2} + 4

= \int_{3}^{8} \frac{x}{( x + 3 ) ^{2} + 4} d x

u置換積分法を適用する

= \int_{6}^{11} \frac{u - 3}{u ^{2} + 4} d u

拡張

\frac{u - 3}{u ^{2} + 4} : \frac{u}{u ^{2} + 4} - \frac{3}{u ^{2} + 4}

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_{6}^{11} \frac{u}{u ^{2} + 4} d u - \int_{6}^{11} \frac{3}{u ^{2} + 4} d u

\int_{6}^{11} \frac{u}{u ^{2} + 4} d u = \frac{1}{2} ln (\frac{25}{8})

\int_{6}^{11} \frac{3}{u ^{2} + 4} d u = \frac{3}{2} (arctan (\frac{11}{2}) - arctan (3))

= \frac{1}{2} ln (\frac{25}{8}) - \frac{3}{2} (arctan (\frac{11}{2}) - arctan (3))

簡素化

\frac{1}{2} ln (\frac{25}{8}) - \frac{3}{2} (arctan (\frac{11}{2}) - arctan (3)) : \frac{ln ( \frac{25}{8} ) - 3 ( arctan ( \frac{11}{2} ) - arctan ( 3 ) )}{2}

= \frac{ln ( \frac{25}{8} ) - 3 ( arctan ( \frac{11}{2} ) - arctan ( 3 ) )}{2}

グラフ

人気の例

(12y^2t^2+10y)dy+(8y^3t)dt=0

(12 y^{2} t^{2} + 10 y) d y + (8 y^{3} t) d t = 0

derivative f(x)=sqrt(11x)

d er i v a t i v e f (x) = 11 x

y^{''}-2y^'+5y=e^xcos(2x)

y^{^{''}} - 2 y^{^{'}} + 5 y = e^{x} cos (2 x)

derivative of cos(3x+1)

\frac{d}{d x} (cos (3 x + 1))

derivative of xsqrt(3-x)

\frac{d}{d x} (x 3 - x)