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受欢迎的微积分 >

integral of-1sin^3(x)cos^2(x)

解答

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

解答

\frac{1}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x) + C

求解步骤

\int - 1 \cdot sin^{3} (x) cos^{2} (x) d x

提出常数:

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

= - \int sin^{3} (x) cos^{2} (x) d x

使用三角恒等式改写

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

使用换元积分法

= - \int - u^{2} (1 - u^{2}) d u

乘开

- u^{2} (1 - u^{2}) : - u^{2} + u^{4}

= - \int - u^{2} + u^{4} d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- \int u^{2} d u + \int u^{4} d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int u^{4} d u = \frac{u ^{5}}{5}

= - (- \frac{u ^{3}}{3} + \frac{u ^{5}}{5})

u = cos (x)

代回

= - (- \frac{cos ^{3} ( x )}{3} + \frac{cos ^{5} ( x )}{5})

化简

- (- \frac{cos ^{3} ( x )}{3} + \frac{cos ^{5} ( x )}{5}) : \frac{1}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x)

= \frac{1}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x)

解答补常数

= \frac{1}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x) + C

作图

流行的例子

面积 x^2+1,3-x,x=0

a re a x^{2} + 1, 3 - x, x = 0

derivative of ((tan(x)/x)^2)

\frac{d}{d x} ((\frac{tan ( x )}{x})^{2})

sum from n=1 to infinity of nln(2)

n = 1 \sum \infty n ln (2)

integral of sin^2(v)cos^3(v)

\int sin^{2} (v) cos^{3} (v) d v

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