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人気のある微分積分 >

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

解

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

解

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

+1

十進法表記

0.96445 \dots

解答ステップ

\int_{1}^{2} \frac{7}{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

= 7 \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 )}

= 7 \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

= 7 (\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))

= 7 (\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}

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

グラフ

人気の例

y^{'''}+12y^{''}+36y^'=18x^2

y^{^{'''}} + 12 y^{^{''}} + 36 y^{^{'}} = 18 x^{2}

derivative x^3+3

d er i v a t i v e x^{3} + 3

derivative f(x)=(3+x^5)^{2/3}

d er i v a t i v e f (x) = (3 + x^{5})^{\frac{2}{3}}

derivative x^3cos(4x)

d er i v a t i v e x^{3} cos (4 x)

integral of (tan^3(x))/(sec^6(x))

\int \frac{tan ^{3} ( x )}{sec ^{6} ( x )} d x