integral of (cos(θ)-cos(2θ))/(1-cos(θ))

解答

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

θ + 2 sin (θ) + C

求解步骤

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

使用三角恒等式改写

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

乘开

\frac{- cos ^{2} ( θ ) + sin ^{2} ( θ ) + cos ( θ )}{1 - cos ( θ )} : - \frac{cos ^{2} ( θ )}{1 - cos ( θ )} + \frac{sin ^{2} ( θ )}{1 - cos ( θ )} + \frac{cos ( θ )}{1 - cos ( θ )}

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

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

\int \frac{cos ^{2} ( θ )}{1 - cos ( θ )} d θ = - cot (\frac{θ}{2}) - sin (θ) - θ

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

\int \frac{cos ( θ )}{1 - cos ( θ )} d θ = - cot (\frac{θ}{2}) - θ

= - (- cot (\frac{θ}{2}) - sin (θ) - θ) + sin (θ) + θ - cot (\frac{θ}{2}) - θ

化简

- (- cot (\frac{θ}{2}) - sin (θ) - θ) + sin (θ) + θ - cot (\frac{θ}{2}) - θ : θ + 2 sin (θ)

= θ + 2 sin (θ)

解答补常数

= θ + 2 sin (θ) + C

d er i v a t i v e x (ln (x))

\frac{\partial}{\partial x} (x + y - 8)

d er i v a t i v e f (x) = (x^{2} + 5) (8 - x)

x \to 1 lim (4 x^{2} + 2)

d er i v a t i v e \frac{4}{x} - 1