What is the integral of sin(2u) ?

The integral of sin(2u) is -1/2 cos(2u)+C

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integral of sin(2u)

\int sin (2 u) d u

- \frac{1}{2} cos (2 u) + C

Solution steps

\int sin (2 u) d u

Apply u-substitution

: \int sin (v) \frac{1}{2} d v

= \int sin (v) \frac{1}{2} d v

Take the constant out:

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

= \frac{1}{2} \cdot \int sin (v) d v

Use the common integral:

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

= \frac{1}{2} (- cos (v))

Substitute back

v = 2 u

= \frac{1}{2} (- cos (2 u))

Simplify

= - \frac{1}{2} cos (2 u)

Add a constant to the solution

= - \frac{1}{2} cos (2 u) + C

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y^{^{'}} + \frac{1}{x} y = \frac{2}{3} x^{2} y^{2}

\int sec^{6} (x) d x

x \to π + lim (cot (x))