What is the integral of (sin(sqrt(x))*cos(sqrt(x)))/(sqrt(x)) ?

The integral of (sin(sqrt(x))*cos(sqrt(x)))/(sqrt(x)) is -1/2 cos(2sqrt(x))+C

integral of (sin(sqrt(x))*cos(sqrt(x)))/(sqrt(x))

\int \frac{sin ( x ) \cdot cos ( x )}{x} d x

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

Solution steps

\int \frac{sin ( x ) cos ( x )}{x} d x

Rewrite using trig identities

= \int \frac{sin ( 2 x )}{2 x} 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 \frac{sin ( 2 x )}{x} d x

Apply u-substitution

= \frac{1}{2} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{2} (- cos (u))

Substitute back

u = 2 x

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

Simplify

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

Add a constant to the solution

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

\int \frac{1}{- 196.2 - 4 v ^{2}} d v

x \to \infty lim (\frac{4}{x})

\frac{d}{d x} ((sec (x))^{3})

\frac{d}{d x} (d cos (t) + t^{2} sin (t))

\frac{d x}{d t} = x^{2} + \frac{1}{25}

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