integral of (4+cos(x))^2

解答

\int (4 + cos (x))^{2} d x

\frac{33}{2} x + \frac{1}{4} sin (2 x) + 8 sin (x) + C

求解步骤

\int (4 + cos (x))^{2} d x

乘开

(4 + cos (x))^{2} : 16 + 8 cos (x) + cos^{2} (x)

= \int 16 + 8 cos (x) + cos^{2} (x) d x

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 16 d x + \int 8 cos (x) d x + \int cos^{2} (x) d x

\int 16 d x = 16 x

\int 8 cos (x) d x = 8 sin (x)

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

= 16 x + 8 sin (x) + \frac{1}{2} (x + \frac{1}{2} sin (2 x))

化简

16 x + 8 sin (x) + \frac{1}{2} (x + \frac{1}{2} sin (2 x)) : \frac{33}{2} x + \frac{1}{4} sin (2 x) + 8 sin (x)

= \frac{33}{2} x + \frac{1}{4} sin (2 x) + 8 sin (x)

解答补常数

= \frac{33}{2} x + \frac{1}{4} sin (2 x) + 8 sin (x) + C

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