What is the integral of (cos^3(sqrt(x)))/(2sqrt(x)) ?

The integral of (cos^3(sqrt(x)))/(2sqrt(x)) is sin(sqrt(x))-1/3 sin^3(sqrt(x))+C

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integral of (cos^3(sqrt(x)))/(2sqrt(x))

\int \frac{cos ^{3} ( x )}{2 x} d x

sin (x) - \frac{1}{3} sin^{3} (x) + C

Solution steps

\int \frac{cos ^{3} ( 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{cos ^{3} ( x )}{x} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

: \int 1 - v^{2} d v

= \frac{1}{2} \cdot 2 \cdot \int 1 - v^{2} d v

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} \cdot 2 (\int 1 d v - \int v^{2} d v)

\int 1 d v = v

\int v^{2} d v = \frac{v ^{3}}{3}

= \frac{1}{2} \cdot 2 (v - \frac{v ^{3}}{3})

Substitute back

= \frac{1}{2} \cdot 2 (sin (x) - \frac{sin ^{3} ( x )}{3})

Simplify

\frac{1}{2} \cdot 2 (sin (x) - \frac{sin ^{3} ( x )}{3}) : sin (x) - \frac{1}{3} sin^{3} (x)

= sin (x) - \frac{1}{3} sin^{3} (x)

Add a constant to the solution

= sin (x) - \frac{1}{3} sin^{3} (x) + C

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What is the integral of (cos^3(sqrt(x)))/(2sqrt(x)) ?
The integral of (cos^3(sqrt(x)))/(2sqrt(x)) is sin(sqrt(x))-1/3 sin^3(sqrt(x))+C