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人気のある微分積分 >

integral of (x^2)/(sqrt(1-2x-x^2))

解

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

解

- \frac{1}{2} sin (2 arcsin (\frac{1}{2} (x + 1))) + 2 arcsin (\frac{1}{2} (x + 1)) + 2 - x^{2} - 2 x + 1 + C

解答ステップ

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

平方完成する

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

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

u置換積分法を適用する

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

拡張

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

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

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

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

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

= arcsin (\frac{1}{2} u) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} u)) - (- 2 - u^{2} + 2) + arcsin (\frac{1}{2} u)

代用を戻す

u = x + 1

= arcsin (\frac{1}{2} (x + 1)) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} (x + 1))) - (- 2 - (x + 1)^{2} + 2) + arcsin (\frac{1}{2} (x + 1))

簡素化

arcsin (\frac{1}{2} (x + 1)) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} (x + 1))) - (- 2 - (x + 1)^{2} + 2) + arcsin (\frac{1}{2} (x + 1)) : - \frac{1}{2} sin (2 arcsin (\frac{1}{2} (x + 1))) + 2 arcsin (\frac{1}{2} (x + 1)) + 2 - x^{2} - 2 x + 1

= - \frac{1}{2} sin (2 arcsin (\frac{1}{2} (x + 1))) + 2 arcsin (\frac{1}{2} (x + 1)) + 2 - x^{2} - 2 x + 1

定数を解答に追加する

= - \frac{1}{2} sin (2 arcsin (\frac{1}{2} (x + 1))) + 2 arcsin (\frac{1}{2} (x + 1)) + 2 - x^{2} - 2 x + 1 + C

グラフ

人気の例

integral of sqrt(2x^2+9)

\int 2 x^{2} + 9 d x

integral of 3/(x^2-4x+1)

\int \frac{3}{x ^{2} - 4 x + 1} d x

integral of (x+1)/(x^2+4x+5)

\int \frac{x + 1}{x ^{2} + 4 x + 5} d x

integral of-2x^5e^{x^6}

\int - 2 x^{5} e^{x^{6}} d x

integral of 1/(2xsqrt(ln(x)))

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