What is the integral of cos(x)(1-sin(x)) ?

The integral of cos(x)(1-sin(x)) is -2cos^2(x/2)(-tan(x/2)+sin^2(x/2))+C

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integral of cos(x)(1-sin(x))

\int cos (x) (1 - sin (x)) d x

- 2 cos^{2} (\frac{x}{2}) (- tan (\frac{x}{2}) + sin^{2} (\frac{x}{2})) + C

Solution steps

\int cos (x) (1 - sin (x)) d x

Multiply by the conjugate of

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

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

Simplify

cos (x) \frac{cos ^{2} ( x )}{1 + sin ( x )} : \frac{cos ^{3} ( x )}{1 + sin ( x )}

= \int \frac{cos ^{3} ( x )}{1 + sin ( x )} d x

Apply u-substitution

: \int - \frac{2 ( u + 1 ) ( u - 1 ) ^{3}}{( u ^{2} + 1 ) ^{3}} d u

= \int - \frac{2 ( u + 1 ) ( u - 1 ) ^{3}}{( u ^{2} + 1 ) ^{3}} d u

Take the constant out:

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

= - 2 \cdot \int \frac{( u + 1 ) ( u - 1 ) ^{3}}{( u ^{2} + 1 ) ^{3}} d u

Take the partial fraction of

\frac{( u + 1 ) ( u - 1 ) ^{3}}{( u ^{2} + 1 ) ^{3}} : \frac{1}{u ^{2} + 1} + \frac{- 2 u - 2}{( u ^{2} + 1 ) ^{2}} + \frac{4 u}{( u ^{2} + 1 ) ^{3}}

= - 2 \cdot \int \frac{1}{u ^{2} + 1} + \frac{- 2 u - 2}{( u ^{2} + 1 ) ^{2}} + \frac{4 u}{( u ^{2} + 1 ) ^{3}} 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

= - 2 (\int \frac{1}{u ^{2} + 1} d u + \int \frac{- 2 u - 2}{( u ^{2} + 1 ) ^{2}} d u + \int \frac{4 u}{( u ^{2} + 1 ) ^{3}} d u)

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

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

\int \frac{4 u}{( u ^{2} + 1 ) ^{3}} d u = - \frac{1}{( u ^{2} + 1 ) ^{2}}

= - 2 (arctan (u) + \frac{1}{u ^{2} + 1} - arctan (u) - \frac{u}{1 + u ^{2}} - \frac{1}{( u ^{2} + 1 ) ^{2}})

Substitute back

u = tan (\frac{x}{2})

= - 2 (arctan (tan (\frac{x}{2})) + \frac{1}{tan ^{2} ( \frac{x}{2} ) + 1} - arctan (tan (\frac{x}{2})) - \frac{tan ( \frac{x}{2} )}{1 + tan ^{2} ( \frac{x}{2} )} - \frac{1}{( tan ^{2} ( \frac{x}{2} ) + 1 ) ^{2}})

Simplify

- 2 (arctan (tan (\frac{x}{2})) + \frac{1}{tan ^{2} ( \frac{x}{2} ) + 1} - arctan (tan (\frac{x}{2})) - \frac{tan ( \frac{x}{2} )}{1 + tan ^{2} ( \frac{x}{2} )} - \frac{1}{( tan ^{2} ( \frac{x}{2} ) + 1 ) ^{2}}) : - 2 cos^{2} (\frac{x}{2}) (- tan (\frac{x}{2}) + sin^{2} (\frac{x}{2}))

= - 2 cos^{2} (\frac{x}{2}) (- tan (\frac{x}{2}) + sin^{2} (\frac{x}{2}))

Add a constant to the solution

= - 2 cos^{2} (\frac{x}{2}) (- tan (\frac{x}{2}) + sin^{2} (\frac{x}{2})) + C

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What is the integral of cos(x)(1-sin(x)) ?
The integral of cos(x)(1-sin(x)) is -2cos^2(x/2)(-tan(x/2)+sin^2(x/2))+C