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人気のある微分積分 >

integral of (x+4)/(x^2+9x-10)

解

\int \frac{x + 4}{x ^{2} + 9 x - 10} d x

解

ln ∣ x + 10 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x + 10 ∣) + C

解答ステップ

\int \frac{x + 4}{x ^{2} + 9 x - 10} d x

因数

x^{2} + 9 x - 10 : (x - 1) (x + 10)

= \int \frac{x + 4}{( x - 1 ) ( x + 10 )} d x

u置換積分法を適用する

= \int \frac{u + 5}{u ( u + 11 )} d u

拡張

\frac{u + 5}{u ( u + 11 )} : \frac{u}{u ( u + 11 )} + \frac{5}{u ( u + 11 )}

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 \frac{u}{u ( u + 11 )} d u + \int \frac{5}{u ( u + 11 )} d u

\int \frac{u}{u ( u + 11 )} d u = ln ∣ u + 11 ∣

\int \frac{5}{u ( u + 11 )} d u = 5 (\frac{1}{11} ln ∣ u ∣ - \frac{1}{11} ln ∣ u + 11 ∣)

= ln ∣ u + 11 ∣ + 5 (\frac{1}{11} ln ∣ u ∣ - \frac{1}{11} ln ∣ u + 11 ∣)

代用を戻す

u = x - 1

= ln ∣ x - 1 + 11 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x - 1 + 11 ∣)

簡素化

ln ∣ x - 1 + 11 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x - 1 + 11 ∣) : ln ∣ x + 10 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x + 10 ∣)

= ln ∣ x + 10 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x + 10 ∣)

定数を解答に追加する

= ln ∣ x + 10 ∣ + 5 (\frac{1}{11} ln ∣ x - 1 ∣ - \frac{1}{11} ln ∣ x + 10 ∣) + C

グラフ

人気の例

derivative f(x)=-2/(x^3)

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

sum from n=0 to infinity of (n/(n+2))^2

n = 0 \sum \infty (\frac{n}{n + 2})^{2}

integral of x\sqrt[8]{64+x^2}

\int x 8 64 + x^{2} d x

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integral of 1/(w^5)

\int \frac{1}{w ^{5}} d w