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受欢迎的微积分 >

integral of (tan(2x)+cot(2x))^2

解答

\int (tan (2 x) + cot (2 x))^{2} d x

解答

\frac{1}{2} tan (2 x) - \frac{1}{2} cot (2 x) + C

求解步骤

\int (tan (2 x) + cot (2 x))^{2} d x

使用换元积分法

= \int \frac{1}{2} tan^{2} (u) + tan (u) cot (u) + \frac{1}{2} cot^{2} (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}{2} tan^{2} (u) d u + \int tan (u) cot (u) d u + \int \frac{1}{2} cot^{2} (u) d u

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

\int tan (u) cot (u) d u = u

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

= \frac{1}{2} (- u + tan (u)) + u + \frac{1}{2} (- u - cot (u))

u = 2 x

代回

= \frac{1}{2} (- 2 x + tan (2 x)) + 2 x + \frac{1}{2} (- 2 x - cot (2 x))

化简

\frac{1}{2} (- 2 x + tan (2 x)) + 2 x + \frac{1}{2} (- 2 x - cot (2 x)) : \frac{1}{2} tan (2 x) - \frac{1}{2} cot (2 x)

= \frac{1}{2} tan (2 x) - \frac{1}{2} cot (2 x)

解答补常数

= \frac{1}{2} tan (2 x) - \frac{1}{2} cot (2 x) + C

作图

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