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受欢迎的微积分 >

integral of-cos^3(x)

解答

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

解答

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

求解步骤

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

提出常数:

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

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

使用三角恒等式改写

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

使用换元积分法

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

\int 1 d u = u

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

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

u = sin (x)

代回

= - (sin (x) - \frac{sin ^{3} ( x )}{3})

化简

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

= - sin (x) + \frac{1}{3} sin^{3} (x)

解答补常数

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

作图

流行的例子

derivative of sin(3x^2cos(2x))

\frac{d}{d x} (sin (3) x^{2} cos (2 x))

integral of t*e^t

\int t \cdot e^{t} d t

y^{''}-y^'+121y=11sin(11t)

y^{^{''}} - y^{^{'}} + 121 y = 11 sin (11 t)

integral of 1/(xsqrt(x-2))

\int \frac{1}{x x - 2} d x

y^{''}+4y^'+4y=e^{2x}

y^{^{''}} + 4 y^{^{'}} + 4 y = e^{2 x}