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受欢迎的微积分 >

integral of 1/(sqrt(16+4x-2x^2))

解答

\int \frac{1}{16 + 4 x - 2 x ^{2}} d x

解答

\frac{1}{2} arcsin (\frac{x - 1}{3}) + C

求解步骤

\int \frac{1}{16 + 4 x - 2 x ^{2}} d x

对

16 + 4 x - 2 x^{2}

配方:

- 2 (x - 1)^{2} + 18

= \int \frac{1}{- 2 ( x - 1 ) ^{2} + 18} d x

使用换元积分法

= \int \frac{1}{- 2 u ^{2} + 18} d u

使用三角换元法

= \int \frac{1}{2} d v

提出常数:

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

= \frac{1}{2} \cdot \int 1 d v

常数积分:

\int a d x = a x

= \frac{1}{2} \cdot 1 \cdot v

代回

= \frac{1}{2} \cdot 1 \cdot arcsin (\frac{2}{3 2} (x - 1))

化简

\frac{1}{2} \cdot 1 \cdot arcsin (\frac{2}{3 2} (x - 1)) : \frac{1}{2} arcsin (\frac{x - 1}{3})

= \frac{1}{2} arcsin (\frac{x - 1}{3})

解答补常数

= \frac{1}{2} arcsin (\frac{x - 1}{3}) + C

作图

流行的例子

integral from 1 to 6 of 2pix(-3/2 x+9)

\int_{1}^{6} 2 π x (- \frac{3}{2} x + 9) d x

tangent (sqrt(x)+1)/(sqrt(x)+2)

t an g e n t \frac{x + 1}{x + 2}

derivative of sqrt(y-x^2)

\frac{d}{d x} (y - x^{2})

integral of (48x)/(4x^2+e)

\int \frac{48 x}{4 x ^{2} + e} d x

integral from-1 to 1 of x/(x^2+1)

\int_{- 1}^{1} \frac{x}{x ^{2} + 1} d x