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

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

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integral of-2cos(2x)

\int - 2 cos (2 x) d x

- sin (2 x) + C

Solution steps

\int - 2 cos (2 x) d x

Take the constant out:

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

= - 2 \cdot \int cos (2 x) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = 2 x

= - 2 \cdot \frac{1}{2} sin (2 x)

Simplify

- 2 \cdot \frac{1}{2} sin (2 x) : - sin (2 x)

= - sin (2 x)

Add a constant to the solution

= - sin (2 x) + C

x \to \frac{4}{π} lim (cos (x))

\frac{d}{d x} (\frac{1}{8} x^{8} - x^{4})

ln (x^{2} - 4 x)

(arcsin (\frac{2 x}{1 + x ^{2}}))^{^{'}}