What is the integral of sin(6t) ?

The integral of sin(6t) is -1/6 cos(6t)+C

integral of sin(6t)

\int sin (6 t) d t

- \frac{1}{6} cos (6 t) + C

Solution steps

\int sin (6 t) d t

Apply u-substitution

= \int sin (u) \frac{1}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{6} (- cos (u))

Substitute back

u = 6 t

= \frac{1}{6} (- cos (6 t))

Simplify

= - \frac{1}{6} cos (6 t)

Add a constant to the solution

= - \frac{1}{6} cos (6 t) + C

\frac{d}{d x} (\frac{sin ( 0.5 x )}{x})

\int - \frac{2000 x}{( 1 + x ^{2} )} d x

t an g e n t x^{2}, at x = 2

\frac{\partial}{\partial x} (3 x y^{2} + 20 x^{2} y^{3})

y^{^{''}} + 2 y^{^{'}} + 2 y = 0, y (\frac{π}{4}) = 2, y^{^{'}} (\frac{π}{4}) = - 2