What is the integral of pi/2 sin(pi/6 t) ?

The integral of pi/2 sin(pi/6 t) is -3cos(pi/6 t)+C

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integral of pi/2 sin(pi/6 t)

\int \frac{π}{2} sin (\frac{π}{6} t) d t

- 3 cos (\frac{π}{6} t) + C

Solution steps

\int \frac{π}{2} sin (\frac{π}{6} t) d t

Take the constant out:

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

= \frac{π}{2} \cdot \int sin (\frac{π}{6} t) d t

Apply u-substitution

: \int sin (u) \frac{6}{π} d u

= \frac{π}{2} \cdot \int sin (u) \frac{6}{π} d u

Take the constant out:

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

= \frac{π}{2} \cdot \frac{6}{π} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{π}{2} \cdot \frac{6}{π} (- cos (u))

Substitute back

u = \frac{π}{6} t

= \frac{π}{2} \cdot \frac{6}{π} (- cos (\frac{π}{6} t))

Simplify

\frac{π}{2} \cdot \frac{6}{π} (- cos (\frac{π}{6} t)) : - 3 cos (\frac{π}{6} t)

= - 3 cos (\frac{π}{6} t)

Add a constant to the solution

= - 3 cos (\frac{π}{6} t) + C

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What is the integral of pi/2 sin(pi/6 t) ?
The integral of pi/2 sin(pi/6 t) is -3cos(pi/6 t)+C