What is the integral of-sin(2x) ?

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

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integral of-sin(2x)

\int - sin (2 x) d x

\frac{1}{2} cos (2 x) + C

Solution steps

\int - sin (2 x) d x

Take the constant out:

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

= - \int sin (2 x) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = 2 x

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

Simplify

= \frac{1}{2} cos (2 x)

Add a constant to the solution

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

\int \frac{4 x}{x + 3} d x

\int x (x + 1) d x

\int_{1}^{100} \frac{1}{x ^{2}} d x

\frac{d y}{d x}, 2 x = y^{4}

x \to 0 lim (\frac{x}{- 4 x ^{3} - 8 x})