What is the integral of 1/2 (11cos(x)-10)^2 ?

The integral of 1/2 (11cos(x)-10)^2 is -1/8 (-121sin(2arcsin(cos(x)))+642arcsin(cos(x))+880sin(x))+C

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

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

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

Solution steps

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

Take the constant out:

\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

Factor

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

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

Simplify

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

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

Apply u-substitution

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

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

Take the constant out:

\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)

Expand

\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)

Apply the Sum Rule:

\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)))

Substitute back

u = cos (x)

= \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))))

Simplify

\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))

Add a constant to the solution

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

\frac{d}{d x} (2 x^{3} y^{2})

\frac{\partial}{\partial x} (cos (x))

y = x (x - 2)^{3}

x \to \infty lim (\frac{3 x}{5})

\frac{\partial}{\partial y} (2 x^{3} y^{3} + 3 x + 4 y)

What is the integral of 1/2 (11cos(x)-10)^2 ?
The integral of 1/2 (11cos(x)-10)^2 is -1/8 (-121sin(2arcsin(cos(x)))+642arcsin(cos(x))+880sin(x))+C