What is the integral of 5cos(2x) ?

The integral of 5cos(2x) is 5/2 sin(2x)+C

integral of 5cos(2x)

\int 5 cos (2 x) d x

\frac{5}{2} sin (2 x) + C

Solution steps

\int 5 cos (2 x) d x

Take the constant out:

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

= 5 \cdot \int cos (2 x) d x

Apply u-substitution

= 5 \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

= 5 \cdot \frac{1}{2} \cdot \int cos (u) d u

Use the common integral:

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

= 5 \cdot \frac{1}{2} sin (u)

Substitute back

u = 2 x

= 5 \cdot \frac{1}{2} sin (2 x)

Simplify

5 \cdot \frac{1}{2} sin (2 x) : \frac{5}{2} sin (2 x)

= \frac{5}{2} sin (2 x)

Add a constant to the solution

= \frac{5}{2} sin (2 x) + C

y^{^{''}} + 4 y^{^{'}} + 4 = 0, y^{^{'}} (0) = 1

a re a f (x) = cos (x), - \frac{π}{2}, \frac{π}{2}

x y^{^{'}} = y + y^{3}

\frac{d y}{d x} + \frac{y}{x} = x y^{2}

x \to 7 lim (\frac{1}{∣ 7 - x ∣})