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受欢迎的微积分 >

integral of (1+4cot(x))/(4-cot(x))

解答

\int \frac{1 + 4 cot ( x )}{4 - cot ( x )} d x

解答

ln ∣ - 4 + cot (x) ∣ - \frac{1}{2} ln (csc^{2} (x)) + C

求解步骤

\int \frac{1 + 4 cot ( x )}{4 - cot ( x )} d x

使用三角换元法

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

提出常数:

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

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

将

\frac{1 + 4 u}{( 4 - u ) ( 1 + u ^{2} )}

用部份分式展开:

- \frac{1}{u - 4} + \frac{u}{u ^{2} + 1}

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

\int \frac{1}{u - 4} d u = ln ∣ u - 4 ∣

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

= - (- ln ∣ u - 4 ∣ + \frac{1}{2} ln u^{2} + 1)

u = cot (x)

代回

= - (- ln ∣ cot (x) - 4 ∣ + \frac{1}{2} ln cot^{2} (x) + 1)

化简

- (- ln ∣ cot (x) - 4 ∣ + \frac{1}{2} ln cot^{2} (x) + 1) : ln ∣ - 4 + cot (x) ∣ - \frac{1}{2} ln (csc^{2} (x))

= ln ∣ - 4 + cot (x) ∣ - \frac{1}{2} ln (csc^{2} (x))

解答补常数

= ln ∣ - 4 + cot (x) ∣ - \frac{1}{2} ln (csc^{2} (x)) + C

作图

流行的例子

integral of x(2x+3)^7

\int x (2 x + 3)^{7} d x

integral of (cot(t)csc(t))/(csc^7(t))

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

integral of (2x-7)/(x^2-12x+32)

\int \frac{2 x - 7}{x ^{2} - 12 x + 32} d x

integral of ((4x^3))/(x^4+1)

\int \frac{( 4 x ^{3} )}{x ^{4} + 1} d x

integral of (xy^2+3x^2y)

\int (x y^{2} + 3 x^{2} y) d x