What is the integral of ((sin(4x))/4)-((sin^3(x))/4) ?

The integral of ((sin(4x))/4)-((sin^3(x))/4) is -1/16 cos(4x)-1/4 (-cos(x)+(cos^3(x))/3)+C

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integral of ((sin(4x))/4)-((sin^3(x))/4)

\int (\frac{sin ( 4 x )}{4}) - (\frac{sin ^{3} ( x )}{4}) d x

- \frac{1}{16} cos (4 x) - \frac{1}{4} (- cos (x) + \frac{cos ^{3} ( x )}{3}) + C

Solution steps

\int \frac{sin ( 4 x )}{4} - \frac{sin ^{3} ( x )}{4} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{sin ( 4 x )}{4} d x - \int \frac{sin ^{3} ( x )}{4} d x

\int \frac{sin ( 4 x )}{4} d x = - \frac{1}{16} cos (4 x)

\int \frac{sin ^{3} ( x )}{4} d x = \frac{1}{4} (- cos (x) + \frac{cos ^{3} ( x )}{3})

= - \frac{1}{16} cos (4 x) - \frac{1}{4} (- cos (x) + \frac{cos ^{3} ( x )}{3})

Add a constant to the solution

= - \frac{1}{16} cos (4 x) - \frac{1}{4} (- cos (x) + \frac{cos ^{3} ( x )}{3}) + C

n = 0 \sum \infty n (n + 1) - 1

\int cos (x + π) d x

\frac{e ^{- s}}{s ^{2} ( s + 2 )}

\int 1 + e^{2 x} d x

\frac{d}{d x} (25 x^{2} e^{2 x} - 16)

What is the integral of ((sin(4x))/4)-((sin^3(x))/4) ?
The integral of ((sin(4x))/4)-((sin^3(x))/4) is -1/16 cos(4x)-1/4 (-cos(x)+(cos^3(x))/3)+C