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人気のある微分積分 >

integral of sqrt(x^2+2x-7)

解

\int x^{2} + 2 x - 7 d x

解

8 (- ln x^{2} + 2 x - 7 + x + 1 + \frac{3}{2} ln (2) + \frac{1}{16} (x + 1) x^{2} + 2 x - 7 + \frac{1}{4} (2 ln x^{2} + 2 x - 7 + x + 1 - 3 ln (2))) + C

解答ステップ

\int x^{2} + 2 x - 7 d x

平方完成する

x^{2} + 2 x - 7 : (x + 1)^{2} - 8

= \int (x + 1)^{2} - 8 d x

u置換積分法を適用する

= \int u^{2} - 8 d u

三角関数による置換を適用する

= \int 8 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

= 8 \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

= 8 \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

拡張

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

= 8 \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

= 8 (- \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) ∣

= 8 (- 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

= 8 (- ln tan (arcsec (\frac{1}{2 2} (x + 1))) + sec (arcsec (\frac{1}{2 2} (x + 1))) + \frac{1}{2} sec (arcsec (\frac{1}{2 2} (x + 1))) tan (arcsec (\frac{1}{2 2} (x + 1))) + \frac{1}{2} ln tan (arcsec (\frac{1}{2 2} (x + 1))) + sec (arcsec (\frac{1}{2 2} (x + 1))))

簡素化

8 (- ln tan (arcsec (\frac{1}{2 2} (x + 1))) + sec (arcsec (\frac{1}{2 2} (x + 1))) + \frac{1}{2} sec (arcsec (\frac{1}{2 2} (x + 1))) tan (arcsec (\frac{1}{2 2} (x + 1))) + \frac{1}{2} ln tan (arcsec (\frac{1}{2 2} (x + 1))) + sec (arcsec (\frac{1}{2 2} (x + 1)))) : 8 (- ln x^{2} + 2 x - 7 + x + 1 + \frac{3}{2} ln (2) + \frac{1}{16} (x + 1) x^{2} + 2 x - 7 + \frac{1}{4} (2 ln x^{2} + 2 x - 7 + x + 1 - 3 ln (2)))

= 8 (- ln x^{2} + 2 x - 7 + x + 1 + \frac{3}{2} ln (2) + \frac{1}{16} (x + 1) x^{2} + 2 x - 7 + \frac{1}{4} (2 ln x^{2} + 2 x - 7 + x + 1 - 3 ln (2)))

定数を解答に追加する

= 8 (- ln x^{2} + 2 x - 7 + x + 1 + \frac{3}{2} ln (2) + \frac{1}{16} (x + 1) x^{2} + 2 x - 7 + \frac{1}{4} (2 ln x^{2} + 2 x - 7 + x + 1 - 3 ln (2))) + C

グラフ

人気の例

integral of-2/7 xcos(2/3 x)

\int - \frac{2}{7} x cos (\frac{2}{3} x) d x

integral of 1/(x(ln(-x))^3)

\int \frac{1}{x ( ln ( - x ) ) ^{3}} d x

integral of 1/2 sqrt(1-5x^2)

\int \frac{1}{2} 1 - 5 x^{2} d x

integral of x/((x+2)^7)

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

integral of (8x+1)cos(x/4)

\int (8 x + 1) cos (\frac{x}{4}) d x