What is the integral of sin(v/3)cos(v/3) ?

The integral of sin(v/3)cos(v/3) is -3/2 cos^2(v/3)+C

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integral of sin(v/3)cos(v/3)

\int sin (\frac{v}{3}) cos (\frac{v}{3}) d v

- \frac{3}{2} cos^{2} (\frac{v}{3}) + C

Solution steps

\int sin (\frac{v}{3}) cos (\frac{v}{3}) d v

Apply u-substitution

: \int - 3 u d u

= \int - 3 u d u

Take the constant out:

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

= - 3 \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= - 3 \cdot \frac{u ^{2}}{2}

Substitute back

u = cos (\frac{v}{3})

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

Simplify

- 3 \cdot \frac{cos ^{2} ( \frac{v}{3} )}{2} : - \frac{3}{2} cos^{2} (\frac{v}{3})

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

Add a constant to the solution

= - \frac{3}{2} cos^{2} (\frac{v}{3}) + C

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