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受欢迎的微积分 >

integral of 1/(sqrt(tan(x)))

解答

\int \frac{1}{tan ( x )} d x

解答

- 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1))) + C

求解步骤

\int \frac{1}{tan ( x )} d x

使用三角恒等式改写

= \int cot (x) d x

使用三角换元法

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

使用换元积分法

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

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

将

\frac{w ^{2}}{4 + w ^{4}}

用部份分式展开:

- \frac{w}{4 ( w ^{2} + 2 w + 2 )} + \frac{w}{4 ( w ^{2} - 2 w + 2 )}

= - 2 \cdot 2 \cdot \int - \frac{w}{4 ( w ^{2} + 2 w + 2 )} + \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w

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

= - 2 \cdot 2 (- \int \frac{w}{4 ( w ^{2} + 2 w + 2 )} d w + \int \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w)

\int \frac{w}{4 ( w ^{2} + 2 w + 2 )} d w = \frac{1}{8} (ln w^{2} + 2 w + 2 - 2 arctan (w + 1))

\int \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w = \frac{1}{8} (ln w^{2} - 2 w + 2 + 2 arctan (w - 1))

= - 2 \cdot 2 (- \frac{1}{8} (ln w^{2} + 2 w + 2 - 2 arctan (w + 1)) + \frac{1}{8} (ln w^{2} - 2 w + 2 + 2 arctan (w - 1)))

代回

= - 2 \cdot 2 (- \frac{1}{8} (ln (2 cot (x))^{2} + 2 2 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln (2 cot (x))^{2} - 2 2 cot (x) + 2 + 2 arctan (2 cot (x) - 1)))

化简

- 2 \cdot 2 (- \frac{1}{8} (ln (2 cot (x))^{2} + 2 2 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln (2 cot (x))^{2} - 2 2 cot (x) + 2 + 2 arctan (2 cot (x) - 1))) : - 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1)))

= - 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1)))

解答补常数

= - 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1))) + C

作图

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\int_{0}^{0.5} 300 x d x

(\partial)/(\partial x)(x*y^2)

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x \to 0 lim ((1 + \frac{x}{2})^{\frac{1}{x}})

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a re a - x^{2} + 8 x, x^{2} - 4 x

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\frac{d}{d x} (∣ x ∣^{∣ c o s (x) ∣})