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

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

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integral of 4cos(x)sin(x)

\int 4 cos (x) sin (x) d x

2 sin^{2} (x) + C

Solution steps

\int 4 cos (x) sin (x) d x

Take the constant out:

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

= 4 \cdot \int cos (x) sin (x) d x

Apply u-substitution

: \int u d u

= 4 \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = sin (x)

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

Simplify

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

= 2 sin^{2} (x)

Add a constant to the solution

= 2 sin^{2} (x) + C

5 x - \frac{3}{x}

\frac{d}{d x} (5 \cdot cos (x))

4 \frac{d y}{d x} = y^{2}

x \to 0 lim (\frac{( 9 ^{x} - 5 ^{x} )}{x})

y^{^{'}} = y \cdot k (100 - y)

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