What is the integral of x-1/2 cos(x^{1/2}) ?

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

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integral of x-1/2 cos(x^{1/2})

integral

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

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

Solution steps

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

Apply u-substitution

: \int 2 u^{3} - u cos (u) d u

= \int 2 u^{3} - u cos (u) 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

= \int 2 u^{3} d u - \int u cos (u) d u

\int 2 u^{3} d u = \frac{u ^{4}}{2}

\int u cos (u) d u = u sin (u) + cos (u)

= \frac{u ^{4}}{2} - (u sin (u) + cos (u))

Substitute back

u = x^{\frac{1}{2}}

= \frac{( x ^{\frac{1}{2}} ) ^{4}}{2} - (x^{\frac{1}{2}} sin (x^{\frac{1}{2}}) + cos (x^{\frac{1}{2}}))

Simplify

\frac{( x ^{\frac{1}{2}} ) ^{4}}{2} - (x^{\frac{1}{2}} sin (x^{\frac{1}{2}}) + cos (x^{\frac{1}{2}})) : \frac{x ^{2}}{2} - x^{\frac{1}{2}} sin (x^{\frac{1}{2}}) - cos (x^{\frac{1}{2}})

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

Add a constant to the solution

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

- 2 cos (2 x)

\frac{( 16 - x )}{( 4 + x )}

\frac{1}{2 + x ^{2}}

sin^{2} (x) - cos^{2} (x)

\frac{( x )}{( 1 - x )}

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