integral of (sin(x))/(2+5cos(x))

解答

\int \frac{sin ( x )}{2 + 5 cos ( x )} d x

- \frac{1}{5} ln ∣ 2 + 5 cos (x) ∣ + C

求解步骤

\int \frac{sin ( x )}{2 + 5 cos ( x )} d x

使用换元积分法

= \int - \frac{1}{5 u} d u

提出常数:

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

= - \frac{1}{5} \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}{5} ln ∣ u ∣

u = 2 + 5 cos (x)

代回

= - \frac{1}{5} ln ∣ 2 + 5 cos (x) ∣

解答补常数

= - \frac{1}{5} ln ∣ 2 + 5 cos (x) ∣ + C

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