What is the integral of (sin(8x))/8 ?

The integral of (sin(8x))/8 is -1/64 cos(8x)+C

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integral of (sin(8x))/8

\int \frac{sin ( 8 x )}{8} d x

- \frac{1}{64} cos (8 x) + C

Solution steps

\int \frac{sin ( 8 x )}{8} d x

Take the constant out:

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

= \frac{1}{8} \cdot \int sin (8 x) d x

Apply u-substitution

= \frac{1}{8} \cdot \int sin (u) \frac{1}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \frac{1}{8} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{8} \cdot \frac{1}{8} (- cos (u))

Substitute back

u = 8 x

= \frac{1}{8} \cdot \frac{1}{8} (- cos (8 x))

Simplify

\frac{1}{8} \cdot \frac{1}{8} (- cos (8 x)) : - \frac{1}{64} cos (8 x)

= - \frac{1}{64} cos (8 x)

Add a constant to the solution

= - \frac{1}{64} cos (8 x) + C

\frac{d y}{d x} = y e^{- x^{2}}, y (4) = 1

y^{2} \frac{d y}{d x} = 3 x^{2} y^{3} - 6 x^{2}

\int sin^{5} (6 x) d x

l a pl a ce t r an s f or m sin (6 t)

f (x) = cos (7 x)

What is the integral of (sin(8x))/8 ?
The integral of (sin(8x))/8 is -1/64 cos(8x)+C