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人気のある微分積分 >

integral of (100)/(x^3-10x^2)

解

\int \frac{100}{x ^{3} - 10 x ^{2}} d x

解

- ln ∣ x ∣ + \frac{10}{x} + ln ∣ x - 10 ∣ + C

解答ステップ

\int \frac{100}{x ^{3} - 10 x ^{2}} d x

定数を除く:

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

= 100 \cdot \int \frac{1}{x ^{3} - 10 x ^{2}} d x

以下の部分分数を得る:

\frac{1}{x ^{3} - 10 x ^{2}} : - \frac{1}{100 x} - \frac{1}{10 x ^{2}} + \frac{1}{100 ( x - 10 )}

= 100 \cdot \int - \frac{1}{100 x} - \frac{1}{10 x ^{2}} + \frac{1}{100 ( x - 10 )} 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

= 100 (- \int \frac{1}{100 x} d x - \int \frac{1}{10 x ^{2}} d x + \int \frac{1}{100 ( x - 10 )} d x)

\int \frac{1}{100 x} d x = \frac{1}{100} ln ∣ x ∣

\int \frac{1}{10 x ^{2}} d x = - \frac{1}{10 x}

\int \frac{1}{100 ( x - 10 )} d x = \frac{1}{100} ln ∣ x - 10 ∣

= 100 (- \frac{1}{100} ln ∣ x ∣ - (- \frac{1}{10 x}) + \frac{1}{100} ln ∣ x - 10 ∣)

簡素化

100 (- \frac{1}{100} ln ∣ x ∣ - (- \frac{1}{10 x}) + \frac{1}{100} ln ∣ x - 10 ∣) : - ln ∣ x ∣ + \frac{10}{x} + ln ∣ x - 10 ∣

= - ln ∣ x ∣ + \frac{10}{x} + ln ∣ x - 10 ∣

定数を解答に追加する

= - ln ∣ x ∣ + \frac{10}{x} + ln ∣ x - 10 ∣ + C

グラフ

人気の例

integral from-6 to 6 of pi(36-x^2)

\int_{- 6}^{6} π (36 - x^{2}) d x

(dy)/(dx)+1/x y=6x^2

\frac{d y}{d x} + \frac{1}{x} y = 6 x^{2}

tangent f(x)= 1/((1+x^2)),\at x=-1

t an g e n t f (x) = \frac{1}{( 1 + x ^{2} )}, at x = - 1

limit as x approaches-10 of (+)f(x)

x \to - 10 lim ((+) f (x))

(\partial)/(\partial y)(e^{13xy})

\frac{\partial}{\partial y} (e^{13 x y})