What is the integral of 2sin(x/2) ?

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

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integral of 2sin(x/2)

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int sin (u) \cdot 2 d u

= 2 \cdot \int sin (u) \cdot 2 d u

Take the constant out:

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

= 2 \cdot 2 \cdot \int sin (u) d u

Use the common integral:

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

= 2 \cdot 2 (- cos (u))

Substitute back

u = \frac{x}{2}

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

Simplify

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

Add a constant to the solution

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

\int \frac{65 x + 7}{( 8 x + 1 ) ( x - 1 )} d x

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

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

y^{^{''}} - 2 y^{^{'}} + y = 3 e^{2 x}

y = \frac{4}{x}, (- 5, - \frac{4}{5})