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受欢迎的微积分 >

integral of (x-3)/(sqrt(6x-x^2))

解答

\int \frac{x - 3}{6 x - x ^{2}} d x

解答

- - x^{2} + 6 x + C

求解步骤

\int \frac{x - 3}{6 x - x ^{2}} d x

对

6 x - x^{2}

配方:

- (x - 3)^{2} + 9

= \int \frac{x - 3}{- ( x - 3 ) ^{2} + 9} d x

使用换元积分法

= \int \frac{u}{- u ^{2} + 9} d u

使用换元积分法

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

使用幂法则

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

代回

= - \frac{1}{2} \cdot 2 (- (x - 3)^{2} + 9)^{\frac{1}{2}}

化简

- \frac{1}{2} \cdot 2 (- (x - 3)^{2} + 9)^{\frac{1}{2}} : - - x^{2} + 6 x

= - - x^{2} + 6 x

解答补常数

= - - x^{2} + 6 x + C

作图

流行的例子

integral of 7\sqrt[6]{x^5}

\int 7 6 x^{5} d x

derivative of ln(cos(sqrt(x)))

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

integral of (tan^2(x))/(sec^5(x))

\int \frac{tan ^{2} ( x )}{sec ^{5} ( x )} d x

integral of w/(sqrt(16-w^2))

\int \frac{w}{16 - w ^{2}} d w

derivative of x/(sqrt(x^2+7))

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