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受欢迎的微积分 >

integral of 1/(x^2)sin(1/x)cos(1/x)

解答

\int \frac{1}{x ^{2}} sin (\frac{1}{x}) cos (\frac{1}{x}) d x

解答

\frac{1}{4} cos (\frac{2}{x}) + C

求解步骤

\int \frac{1}{x ^{2}} sin (\frac{1}{x}) cos (\frac{1}{x}) d x

使用三角恒等式改写

= \int \frac{sin ( \frac{2}{x} )}{2 x ^{2}} d x

提出常数:

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

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

使用换元积分法

= \frac{1}{2} \cdot \int - \frac{sin ( u )}{2} d u

提出常数:

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

= \frac{1}{2} (- \frac{1}{2} \cdot \int sin (u) d u)

使用积分常见定则:

\int sin (u) d u = - cos (u)

= \frac{1}{2} (- \frac{1}{2} (- cos (u)))

u = \frac{2}{x}

代回

= \frac{1}{2} (- \frac{1}{2} (- cos (\frac{2}{x})))

化简

\frac{1}{2} (- \frac{1}{2} (- cos (\frac{2}{x}))) : \frac{1}{4} cos (\frac{2}{x})

= \frac{1}{4} cos (\frac{2}{x})

解答补常数

= \frac{1}{4} cos (\frac{2}{x}) + C

作图

流行的例子

integral from 0 to 2 of 4x^2-2x^3

\int_{0}^{2} 4 x^{2} - 2 x^{3} d x

derivative of x(x-3^2)

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

面积 y=x^2,y=-x^2+4x

a re a y = x^{2}, y = - x^{2} + 4 x

integral of 12sin^6(x)cos^3(x)

\int 12 sin^{6} (x) cos^{3} (x) d x

derivative f(x)=x^{5/11}(x-5)^2

d er i v a t i v e f (x) = x^{\frac{5}{11}} (x - 5)^{2}