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受欢迎的微积分 >

integral of sin^3(4x)cos^{-2}(4x)

解答

\int sin^{3} (4 x) cos^{- 2} (4 x) d x

解答

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

求解步骤

\int sin^{3} (4 x) cos^{- 2} (4 x) d x

使用换元积分法

= \int \frac{sin ^{3} ( u )}{4 cos ^{2} ( u )} d u

提出常数:

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

= \frac{1}{4} \cdot \int \frac{sin ^{3} ( u )}{cos ^{2} ( u )} d u

使用三角恒等式改写

= \frac{1}{4} \cdot \int \frac{( 1 - cos ^{2} ( u ) ) sin ( u )}{cos ^{2} ( u )} d u

使用换元积分法

= \frac{1}{4} \cdot \int - \frac{1 - v ^{2}}{v ^{2}} d v

乘开

- \frac{1 - v ^{2}}{v ^{2}} : - \frac{1}{v ^{2}} + 1

= \frac{1}{4} \cdot \int - \frac{1}{v ^{2}} + 1 d v

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

= \frac{1}{4} (- \int \frac{1}{v ^{2}} d v + \int 1 d v)

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

\int 1 d v = v

= \frac{1}{4} (- (- \frac{1}{v}) + v)

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

integral of cot^7(x/2)csc^2(x/2)

\int cot^{7} (\frac{x}{2}) csc^{2} (\frac{x}{2}) d x

斜率 (-2,-9),(0,-3)

s l o p e (- 2, - 9), (0, - 3)

derivative of (x-3^3)

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

y^'=(3x^2-1)/(3+2y)

y^{^{'}} = \frac{3 x ^{2} - 1}{3 + 2 y}

derivative of axsin(kx+bxcos(kx))

\frac{d}{d x} (a x sin (k x) + b x cos (k x))