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人気のある微分積分 >

integral from 1 to 2 of (11)/(x^3+3x)

解

\int_{1}^{2} \frac{11}{x ^{3} + 3 x} d x

解

11 \cdot \frac{4 ln ( 2 ) - ln ( 7 )}{6}

+1

十進法表記

1.51557 \dots

解答ステップ

\int_{1}^{2} \frac{11}{x ^{3} + 3 x} d x

定数を除く:

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

= 11 \cdot \int_{1}^{2} \frac{1}{x ^{3} + 3 x} d x

以下の部分分数を得る:

\frac{1}{x ^{3} + 3 x} : \frac{1}{3 x} - \frac{x}{3 ( x ^{2} + 3 )}

= 11 \cdot \int_{1}^{2} \frac{1}{3 x} - \frac{x}{3 ( x ^{2} + 3 )} d x

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

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

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

\int_{1}^{2} \frac{x}{3 ( x ^{2} + 3 )} d x = \frac{1}{6} (ln (7) - 2 ln (2))

= 11 (\frac{1}{3} ln (2) - \frac{1}{6} (ln (7) - 2 ln (2)))

結合

\frac{1}{3} ln (2) - \frac{1}{6} (ln (7) - 2 ln (2)) : \frac{4 ln ( 2 ) - ln ( 7 )}{6}

= 11 \cdot \frac{4 ln ( 2 ) - ln ( 7 )}{6}

グラフ

人気の例

integral of 1/(x^2*sqrt(9+x^2))

\int \frac{1}{x ^{2} \cdot 9 + x ^{2}} d x

integral of x/(sqrt(25+x^2))

\int \frac{x}{25 + x ^{2}} d x

tangent f(x)=2x^3+3x^2-12x+5

t an g e n t f (x) = 2 x^{3} + 3 x^{2} - 12 x + 5

limit as n approaches infinity of n-1

n \to \infty lim (n - 1)

(dy)/(dx)=(x^2)/(y(1+x^3))

\frac{d y}{d x} = \frac{x ^{2}}{y ( 1 + x ^{3} )}