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人気のある微分積分 >

integral of x/(sqrt(x^2-2ax+2a^2))

解

\int \frac{x}{x ^{2} - 2 a x + 2 a ^{2}} d x

解

x^{2} - 2 a x + 2 a^{2} + a ln (\frac{x - a + x ^{2} - 2 a x + 2 a ^{2}}{∣ a ∣}) + C

解答ステップ

\int \frac{x}{x ^{2} - 2 a x + 2 a ^{2}} d x

平方完成する

x^{2} - 2 a x + 2 a^{2} : (x - a)^{2} + a^{2}

= \int \frac{x}{( x - a ) ^{2} + a ^{2}} d x

u置換積分法を適用する

= \int \frac{u + a}{u ^{2} + a ^{2}} d u

拡張

\frac{u + a}{u ^{2} + a ^{2}} : \frac{u}{u ^{2} + a ^{2}} + \frac{a}{u ^{2} + a ^{2}}

= \int \frac{u}{u ^{2} + a ^{2}} + \frac{a}{u ^{2} + a ^{2}} 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

= \int \frac{u}{u ^{2} + a ^{2}} d u + \int \frac{a}{u ^{2} + a ^{2}} d u

\int \frac{u}{u ^{2} + a ^{2}} d u = u^{2} + a^{2}

\int \frac{a}{u ^{2} + a ^{2}} d u = a ln (\frac{u + a ^{2} + u ^{2}}{∣ a ∣})

= u^{2} + a^{2} + a ln (\frac{u + a ^{2} + u ^{2}}{∣ a ∣})

代用を戻す

u = x - a

= (x - a)^{2} + a^{2} + a ln \frac{x - a + a ^{2} + ( x - a ) ^{2}}{∣ a ∣}

簡素化

(x - a)^{2} + a^{2} + a ln \frac{x - a + a ^{2} + ( x - a ) ^{2}}{∣ a ∣} : x^{2} - 2 a x + 2 a^{2} + a ln (\frac{x - a + x ^{2} - 2 a x + 2 a ^{2}}{∣ a ∣})

= x^{2} - 2 a x + 2 a^{2} + a ln (\frac{x - a + x ^{2} - 2 a x + 2 a ^{2}}{∣ a ∣})

定数を解答に追加する

= x^{2} - 2 a x + 2 a^{2} + a ln (\frac{x - a + x ^{2} - 2 a x + 2 a ^{2}}{∣ a ∣}) + C

グラフ

人気の例

integral of (x^3-x^2-12x)/((x-4)(x+3))

\int \frac{x ^{3} - x ^{2} - 12 x}{( x - 4 ) ( x + 3 )} d x

integral of 1/(2+2sin(x)+cos(x))

\int \frac{1}{2 + 2 sin ( x ) + cos ( x )} d x

integral of (x^2)/(x^4+2x^2+1)

\int \frac{x ^{2}}{x ^{4} + 2 x ^{2} + 1} d x

integral of 1/(x(ln(x^2))^7)

\int \frac{1}{x ( ln ( x ^{2} ) ) ^{7}} d x

integral of (x^5)/(sqrt(1-x^3))

\int \frac{x ^{5}}{1 - x ^{3}} d x