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人気のある微分積分 >

integral of 1/(x^2+xsqrt(x))

解

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

解

2 (- ln x - \frac{1}{x} + ln x + 1) + C

解答ステップ

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

u置換積分法を適用する

= \int \frac{2}{u ^{2} ( u + 1 )} d u

定数を除く:

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

= 2 \cdot \int \frac{1}{u ^{2} ( u + 1 )} d u

以下の部分分数を得る:

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

= 2 \cdot \int - \frac{1}{u} + \frac{1}{u ^{2}} + \frac{1}{u + 1} d u

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

= 2 (- \int \frac{1}{u} d u + \int \frac{1}{u ^{2}} d u + \int \frac{1}{u + 1} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

\int \frac{1}{u + 1} d u = ln ∣ u + 1 ∣

= 2 (- ln ∣ u ∣ - \frac{1}{u} + ln ∣ u + 1 ∣)

代用を戻す

u = x

= 2 (- ln x - \frac{1}{x} + ln x + 1)

定数を解答に追加する

= 2 (- ln x - \frac{1}{x} + ln x + 1) + C

グラフ

人気の例

limit as x approaches 7 of 2

x \to 7 lim (2)

tangent f(x)= 5/x ,\at x=3

t an g e n t f (x) = \frac{5}{x}, at x = 3

integral of 36x(9x^2-7)^3

\int 36 x (9 x^{2} - 7)^{3} d x

integral of 5tan(5x)

\int 5 tan (5 x) d x

limit as x approaches 4-of x+1

x \to 4 - lim (x + 1)