What is the integral of 5ysin(4x) ?

The integral of 5ysin(4x) is -5/4 ycos(4x)+C

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integral of 5ysin(4x)

\int 5 y sin (4 x) d x

- \frac{5}{4} y cos (4 x) + C

Solution steps

\int 5 y sin (4 x) d x

Take the constant out:

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

= 5 y \cdot \int sin (4 x) d x

Apply u-substitution

= 5 y \cdot \int sin (u) \frac{1}{4} d u

Take the constant out:

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

= 5 y \frac{1}{4} \cdot \int sin (u) d u

Use the common integral:

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

= 5 y \frac{1}{4} (- cos (u))

Substitute back

u = 4 x

= 5 y \frac{1}{4} (- cos (4 x))

Simplify

5 y \frac{1}{4} (- cos (4 x)) : - \frac{5}{4} y cos (4 x)

= - \frac{5}{4} y cos (4 x)

Add a constant to the solution

= - \frac{5}{4} y cos (4 x) + C

What is the integral of 5ysin(4x) ?
The integral of 5ysin(4x) is -5/4 ycos(4x)+C