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受欢迎的微积分 >

integral of (sin(a)x)/(cos(a)x+b)

解答

\int \frac{sin ( a ) x}{cos ( a ) x + b} d x

解答

sec^{2} (a) sin (a) (x cos (a) + b - b ln ∣ x cos (a) + b ∣) + C

求解步骤

\int \frac{sin ( a ) x}{cos ( a ) x + b} d x

提出常数:

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

= sin (a) \cdot \int \frac{x}{cos ( a ) x + b} d x

使用换元积分法

= sin (a) \cdot \int \frac{u - b}{u cos ^{2} ( a )} d u

提出常数:

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

= sin (a) \frac{1}{cos ^{2} ( a )} \cdot \int \frac{u - b}{u} d u

乘开

\frac{u - b}{u} : 1 - \frac{b}{u}

= sin (a) \frac{1}{cos ^{2} ( a )} \cdot \int 1 - \frac{b}{u} d u

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

= sin (a) \frac{1}{cos ^{2} ( a )} (\int 1 d u - \int \frac{b}{u} d u)

\int 1 d u = u

\int \frac{b}{u} d u = b ln ∣ u ∣

= sin (a) \frac{1}{cos ^{2} ( a )} (u - b ln ∣ u ∣)

u = cos (a) x + b

代回

= sin (a) \frac{1}{cos ^{2} ( a )} (cos (a) x + b - b ln ∣ cos (a) x + b ∣)

使用基本三角恒等式:

\frac{1}{cos ( x )} = sec (x)

= sec^{2} (a) sin (a) (x cos (a) + b - b ln ∣ x cos (a) + b ∣)

解答补常数

= sec^{2} (a) sin (a) (x cos (a) + b - b ln ∣ x cos (a) + b ∣) + C

作图

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