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受欢迎的微积分 >

integral of (cos^5(x))/(sin^3(x))

解答

\int \frac{cos ^{5} ( x )}{sin ^{3} ( x )} d x

解答

- \frac{1}{2} csc^{2} (x) - 2 ln ∣ sin (x) ∣ + \frac{1}{2} sin^{2} (x) + C

求解步骤

\int \frac{cos ^{5} ( x )}{sin ^{3} ( x )} d x

使用三角恒等式改写

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

使用换元积分法

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

乘开

\frac{( 1 - u ^{2} ) ^{2}}{u ^{3}} : \frac{1}{u ^{3}} - \frac{2}{u} + u

= \int \frac{1}{u ^{3}} - \frac{2}{u} + u 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 \frac{1}{u ^{3}} d u - \int \frac{2}{u} d u + \int u d u

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

\int \frac{2}{u} d u = 2 ln ∣ u ∣

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

= - \frac{1}{2 u ^{2}} - 2 ln ∣ u ∣ + \frac{u ^{2}}{2}

u = sin (x)

代回

= - \frac{1}{2 sin ^{2} ( x )} - 2 ln ∣ sin (x) ∣ + \frac{sin ^{2} ( x )}{2}

化简

- \frac{1}{2 sin ^{2} ( x )} - 2 ln ∣ sin (x) ∣ + \frac{sin ^{2} ( x )}{2} : - \frac{1}{2} csc^{2} (x) - 2 ln ∣ sin (x) ∣ + \frac{1}{2} sin^{2} (x)

= - \frac{1}{2} csc^{2} (x) - 2 ln ∣ sin (x) ∣ + \frac{1}{2} sin^{2} (x)

解答补常数

= - \frac{1}{2} csc^{2} (x) - 2 ln ∣ sin (x) ∣ + \frac{1}{2} sin^{2} (x) + C

作图

流行的例子

integral from 0 to 0.6 of 1/((1-x)(3-x))

\int_{0}^{0.6} \frac{1}{( 1 - x ) ( 3 - x )} d x

integral of 1/(t^2sqrt(2-t^2))

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

derivative ((cos(x)))/(cos^2(*x)-1)

d er i v a t i v e \frac{( cos ( x ) )}{cos ^{2} ( \cdot x ) - 1}

integral of (n+3)3

\int (n + 3) 3 d n

integral of 2/(e^x+2)

\int \frac{2}{e ^{x} + 2} d x