What is the integral of cos(x/4)sin(x/4) ?

The integral of cos(x/4)sin(x/4) is -2cos^2(x/4)+C

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integral of cos(x/4)sin(x/4)

\int cos (\frac{x}{4}) sin (\frac{x}{4}) d x

- 2 cos^{2} (\frac{x}{4}) + C

Solution steps

\int cos (\frac{x}{4}) sin (\frac{x}{4}) d x

Apply u-substitution

: \int - 4 u d u

= \int - 4 u d u

Take the constant out:

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

= - 4 \cdot \int u d u

Apply the Power Rule

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

= - 4 \cdot \frac{u ^{2}}{2}

Substitute back

u = cos (\frac{x}{4})

= - 4 \cdot \frac{cos ^{2} ( \frac{x}{4} )}{2}

Simplify

- 4 \cdot \frac{cos ^{2} ( \frac{x}{4} )}{2} : - 2 cos^{2} (\frac{x}{4})

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

Add a constant to the solution

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

arctan (x^{3})

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\int tan (4 x se) c^{2} 4 x d x

What is the integral of cos(x/4)sin(x/4) ?
The integral of cos(x/4)sin(x/4) is -2cos^2(x/4)+C