What is the integral of (1+cos(4t))/2 ?

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

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integral of (1+cos(4t))/2

\int \frac{1 + cos ( 4 t )}{2} d t

\frac{1}{2} (t + \frac{1}{4} sin (4 t)) + C

Solution steps

\int \frac{1 + cos ( 4 t )}{2} d t

Take the constant out:

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

= \frac{1}{2} \cdot \int 1 + cos (4 t) d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int 1 d t + \int cos (4 t) d t)

\int 1 d t = t

\int cos (4 t) d t = \frac{1}{4} sin (4 t)

= \frac{1}{2} (t + \frac{1}{4} sin (4 t))

Add a constant to the solution

= \frac{1}{2} (t + \frac{1}{4} sin (4 t)) + C

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