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

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

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integral of 4cos(2t)

\int 4 cos (2 t) d t

2 sin (2 t) + C

Solution steps

\int 4 cos (2 t) d t

Take the constant out:

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

= 4 \cdot \int cos (2 t) d t

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

= 4 \cdot \frac{1}{2} sin (u)

Substitute back

u = 2 t

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

Simplify

4 \cdot \frac{1}{2} sin (2 t) : 2 sin (2 t)

= 2 sin (2 t)

Add a constant to the solution

= 2 sin (2 t) + C