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受欢迎的微积分 >

integral of 1/(sqrt(x)(\sqrt{x-2))^3}

解答

\int \frac{1}{x ( x - 2 ) ^{3}} d x

解答

- \frac{x}{x - 2} + C

求解步骤

\int \frac{1}{x ( x - 2 ) ^{3}} d x

使用换元积分法

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

提出常数:

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

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

使用三角换元法

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

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{sec ( v )}{tan ^{2} ( v )} d v

使用三角恒等式改写

= 2 \cdot \frac{1}{2} \cdot \int \frac{cos ( v )}{sin ^{2} ( v )} d v

使用换元积分法

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{w ^{2}} d w

使用幂法则

= 2 \cdot \frac{1}{2} (- \frac{1}{w})

代回

= 2 \cdot \frac{1}{2} - \frac{1}{sin ( arctan ( \frac{1}{2} x - 2 ) )}

化简

2 \cdot \frac{1}{2} - \frac{1}{sin ( arctan ( \frac{1}{2} x - 2 ) )} : - \frac{x}{x - 2}

= - \frac{x}{x - 2}

解答补常数

= - \frac{x}{x - 2} + C

作图

流行的例子

integral of (tan(x)+sin(x))cos(x)

\int (tan (x) + sin (x)) cos (x) d x

integral of 15sin^4(3x)cos(3x)

\int 15 sin^{4} (3 x) cos (3 x) d x

integral of x^7-6x+8

\int x^{7} - 6 x + 8 d x

integral of 2/(1+4x^2)+ln(8)*8^x

\int \frac{2}{1 + 4 x ^{2}} + ln (8) \cdot 8^{x} d x

integral of-1/2 y^2

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