What is the integral of (sin(1+2x))/2 ?

The integral of (sin(1+2x))/2 is -1/4 cos(1+2x)+C

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integral of (sin(1+2x))/2

\int \frac{sin ( 1 + 2 x )}{2} d x

- \frac{1}{4} cos (1 + 2 x) + C

Solution steps

\int \frac{sin ( 1 + 2 x )}{2} 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 sin (1 + 2 x) d x

Apply u-substitution

: \int sin (u) \frac{1}{2} d u

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{2} \cdot \frac{1}{2} (- cos (u))

Substitute back

u = 1 + 2 x

= \frac{1}{2} \cdot \frac{1}{2} (- cos (1 + 2 x))

Simplify

\frac{1}{2} \cdot \frac{1}{2} (- cos (1 + 2 x)) : - \frac{1}{4} cos (1 + 2 x)

= - \frac{1}{4} cos (1 + 2 x)

Add a constant to the solution

= - \frac{1}{4} cos (1 + 2 x) + C

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What is the integral of (sin(1+2x))/2 ?
The integral of (sin(1+2x))/2 is -1/4 cos(1+2x)+C