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受欢迎的微积分 >

integral of 1/(sqrt(1-cos(2x)))

解答

\int \frac{1}{1 - cos ( 2 x )} d x

解答

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

求解步骤

\int \frac{1}{1 - cos ( 2 x )} d x

使用三角恒等式改写

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

sin^{2} (x) = (sin (x)),

假定

sin (x) \geq 0

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

使用换元积分法

= \frac{1}{2} \cdot \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

u = tan (\frac{x}{2})

代回

= \frac{1}{2} ln tan (\frac{x}{2})

解答补常数

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

作图

流行的例子

integral of 1/((2x-3)sqrt(5-12x+4x^2))

\int \frac{1}{( 2 x - 3 ) 5 - 12 x + 4 x ^{2}} d x

integral of (t-2/t)(t+2/t)

\int (t - \frac{2}{t}) (t + \frac{2}{t}) d t

integral of x/((sqrt(9+x^2))^3)

\int \frac{x}{( 9 + x ^{2} ) ^{3}} d x

integral of 1/((1+y)sqrt(y))

\int \frac{1}{( 1 + y ) y} d y

integral of-(12e^{3x})/(e^{3x)+4}

\int - \frac{12 e ^{3 x}}{e ^{3 x} + 4} d x