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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

平方完成する

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

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

u置換積分法を適用する

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

三角関数による置換を適用する

= \int 2 tan^{2} (v) sec (v) d v

定数を除く:

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

= 2 \cdot \int tan^{2} (v) sec (v) d v

三角関数の公式を使用して書き換える

= 2 \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

拡張

(- 1 + sec^{2} (v)) sec (v) : - sec (v) + sec^{3} (v)

= 2 \cdot \int - sec (v) + sec^{3} (v) d v

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 sec (v) d v + \int sec^{3} (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

\int sec^{3} (v) d v = \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

= 2 (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代用を戻す

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

簡素化

2 (- ln tan (arcsec (\frac{1}{2} (x - 1))) + sec (arcsec (\frac{1}{2} (x - 1))) + \frac{1}{2} sec (arcsec (\frac{1}{2} (x - 1))) tan (arcsec (\frac{1}{2} (x - 1))) + \frac{1}{2} ln tan (arcsec (\frac{1}{2} (x - 1))) + sec (arcsec (\frac{1}{2} (x - 1)))) : - ln \frac{1}{2} (x - 1) + \frac{1}{2} (x^{2} - 2 x - 1) + \frac{1}{2} (x - 1) x^{2} - 2 x - 1

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

定数を解答に追加する

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

グラフ

人気の例

integral of (csc^2(sqrt(x)))/(sqrt(x))

\int \frac{csc ^{2} ( x )}{x} d x

integral of (10x^9)/(x^{10)+1}

\int \frac{10 x ^{9}}{x ^{10} + 1} d x

integral of x(x^2+6)^2

\int x (x^{2} + 6)^{2} d x

integral of (ln(x))/(4x)

\int \frac{ln ( x )}{4 x} d x

integral of ((cos(x))/x)

\int (\frac{cos ( x )}{x}) d x