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受欢迎的微积分 >

integral of 1/(2sqrt(tan(x)))

解答

\int \frac{1}{2 tan ( x )} d x

解答

- 2 (- \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}{2 tan ( x )} d x

提出常数:

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

= \frac{1}{2} \cdot \int \frac{1}{tan ( x )} d x

使用三角恒等式改写

= \frac{1}{2} \cdot \int cot (x) d x

使用三角换元法

= \frac{1}{2} \cdot \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

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

使用换元积分法

= \frac{1}{2} (- \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

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

使用换元积分法

= \frac{1}{2} (- 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

= \frac{1}{2} (- 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 )}

= \frac{1}{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

= \frac{1}{2} (- 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))

= \frac{1}{2} (- 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

= \frac{1}{2} (- 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

\frac{1}{2} (- 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)))) : - 2 (- \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)))

= - 2 (- \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

= - 2 (- \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

流行的例子

inverselaplace (s^2+3s+5)/(s^3(s+2))

in v erse l a pl a ce \frac{s ^{2} + 3 s + 5}{s ^{3} ( s + 2 )}

(dy)/(dx)=sqrt(x)

\frac{d y}{d x} = x

sum from n=0 to infinity of 2/(n^2)

n = 0 \sum \infty \frac{2}{n ^{2}}

integral from-4 to 2 of (x+4)

\int_{- 4}^{2} (x + 4) d x

面积 y^2=4x,y=x^2

a re a y^{2} = 4 x, y = x^{2}