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受欢迎的微积分 >

integral of 1/2 (11cos(x)-10)^2

解答

\int \frac{1}{2} (11 cos (x) - 10)^{2} d x

解答

- \frac{1}{8} (- 121 sin (2 arcsin (cos (x))) + 642 arcsin (cos (x)) + 880 sin (x)) + C

求解步骤

\int \frac{1}{2} (11 cos (x) - 10)^{2} d x

提出常数:

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

= \frac{1}{2} \cdot \int (11 cos (x) - 10)^{2} d x

因式分解

11 cos (x) - 10 : - (- 11 cos (x) + 10)

= \frac{1}{2} \cdot \int (- (- 11 cos (x) + 10))^{2} d x

化简

(- (- 11 cos (x) + 10))^{2} : (- 11 cos (x) + 10)^{2}

= \frac{1}{2} \cdot \int (- 11 cos (x) + 10)^{2} d x

使用换元积分法

= \frac{1}{2} \cdot \int - \frac{( - 11 u + 10 ) ^{2}}{1 - u ^{2}} d u

提出常数:

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

= \frac{1}{2} (- \int \frac{( - 11 u + 10 ) ^{2}}{1 - u ^{2}} d u)

乘开

\frac{( - 11 u + 10 ) ^{2}}{1 - u ^{2}} : \frac{121 u ^{2}}{1 - u ^{2}} - \frac{220 u}{1 - u ^{2}} + \frac{100}{1 - u ^{2}}

= \frac{1}{2} (- \int \frac{121 u ^{2}}{1 - u ^{2}} - \frac{220 u}{1 - u ^{2}} + \frac{100}{1 - u ^{2}} 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

= \frac{1}{2} (- (\int \frac{121 u ^{2}}{1 - u ^{2}} d u - \int \frac{220 u}{1 - u ^{2}} d u + \int \frac{100}{1 - u ^{2}} d u))

\int \frac{121 u ^{2}}{1 - u ^{2}} d u = \frac{121}{2} (arcsin (u) - \frac{1}{2} sin (2 arcsin (u)))

\int \frac{220 u}{1 - u ^{2}} d u = - 220 1 - u^{2}

\int \frac{100}{1 - u ^{2}} d u = 100 arcsin (u)

= \frac{1}{2} (- (\frac{121}{2} (arcsin (u) - \frac{1}{2} sin (2 arcsin (u))) - (- 220 1 - u^{2}) + 100 arcsin (u)))

u = cos (x)

代回

= \frac{1}{2} (- (\frac{121}{2} (arcsin (cos (x)) - \frac{1}{2} sin (2 arcsin (cos (x)))) - (- 220 1 - cos^{2} (x)) + 100 arcsin (cos (x))))

化简

\frac{1}{2} (- (\frac{121}{2} (arcsin (cos (x)) - \frac{1}{2} sin (2 arcsin (cos (x)))) - (- 220 1 - cos^{2} (x)) + 100 arcsin (cos (x)))) : - \frac{1}{8} (- 121 sin (2 arcsin (cos (x))) + 642 arcsin (cos (x)) + 880 sin (x))

= - \frac{1}{8} (- 121 sin (2 arcsin (cos (x))) + 642 arcsin (cos (x)) + 880 sin (x))

解答补常数

= - \frac{1}{8} (- 121 sin (2 arcsin (cos (x))) + 642 arcsin (cos (x)) + 880 sin (x)) + C

作图

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