What is the integral of cos(3x+pi/2) ?

The integral of cos(3x+pi/2) is 1/3 cos(3x)+C

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integral of cos(3x+pi/2)

\int cos (3 x + \frac{π}{2}) d x

\frac{1}{3} cos (3 x) + C

Solution steps

\int cos (3 x + \frac{π}{2}) d x

Rewrite using trig identities

= \int - sin (3 x) d x

Take the constant out:

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

= - \int sin (3 x) d x

Apply u-substitution

: \int sin (u) \frac{1}{3} d u

= - \int sin (u) \frac{1}{3} d u

Take the constant out:

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

= - \frac{1}{3} \cdot \int sin (u) d u

Use the common integral:

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

= - \frac{1}{3} (- cos (u))

Substitute back

u = 3 x

= - \frac{1}{3} (- cos (3 x))

Simplify

= \frac{1}{3} cos (3 x)

Add a constant to the solution

= \frac{1}{3} cos (3 x) + C

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What is the integral of cos(3x+pi/2) ?
The integral of cos(3x+pi/2) is 1/3 cos(3x)+C