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受欢迎的微积分 >

integral of (sin(x))/(sqrt(cos(2x)))

解答

\int \frac{sin ( x )}{cos ( 2 x )} d x

解答

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

求解步骤

\int \frac{sin ( x )}{cos ( 2 x )} d x

使用三角恒等式改写

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

使用换元积分法

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

使用换元积分法

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

使用三角换元法

= - \frac{1}{2} \cdot \int sec (w) d w

使用积分常见定则:

\int sec (w) d w = ln ∣ tan (w) + sec (w) ∣

= - \frac{1}{2} ln ∣ tan (w) + sec (w) ∣

代回

= - \frac{1}{2} ln tan (arcsec (\frac{1}{2} \cdot 2 cos (x))) + sec (arcsec (\frac{1}{2} \cdot 2 cos (x)))

化简

- \frac{1}{2} ln tan (arcsec (\frac{1}{2} \cdot 2 cos (x))) + sec (arcsec (\frac{1}{2} \cdot 2 cos (x))) : - \frac{1}{2} ln cos (2 x) + 2 cos (x)

= - \frac{1}{2} ln cos (2 x) + 2 cos (x)

解答补常数

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

作图

流行的例子

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

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

integral of 7/(3x+11)

\int \frac{7}{3 x + 11} d x

integral of (x^2-3)/(sqrt(x))

\int \frac{x ^{2} - 3}{x} d x

integral of 1/(\sqrt[n]{t^m)}

\int \frac{1}{n t ^{m}} d t

integral of (((x+2))/(x^3+2x^2+5x))

\int (\frac{( x + 2 )}{x ^{3} + 2 x ^{2} + 5 x}) d x