integral of cos^2(2aθ)

解答

\int cos^{2} (2 a θ) d θ

θ - \frac{θ}{2} + \frac{1}{8 a} sin (4 a θ) + C

求解步骤

\int cos^{2} (2 a θ) d θ

使用三角恒等式改写

= \int 1 - sin^{2} (2 a θ) d 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 1 d θ - \int sin^{2} (2 a θ) d θ

\int 1 d θ = θ

\int sin^{2} (2 a θ) d θ = \frac{θ}{2} - \frac{1}{8 a} sin (4 a θ)

= θ - (\frac{θ}{2} - \frac{1}{8 a} sin (4 a θ))

化简

= θ - \frac{θ}{2} + \frac{1}{8 a} sin (4 a θ)

解答补常数

= θ - \frac{θ}{2} + \frac{1}{8 a} sin (4 a θ) + C

\frac{\partial}{\partial x} (5 \cdot (x - 4) \cdot y^{3})

\frac{d}{d x} ((2 x - 1)^{\frac{1}{2}})

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

\int (4 x + 5 x) d x

(9 x + 0.8 x^{2} + 0.07 x^{3})^{^{'}}