What is the integral of cos(x)-cos^4(x) ?

The integral of cos(x)-cos^4(x) is sin(x)-cos^3(x)sin(x)-3/8 x+3/32 sin(4x)+C

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integral of cos(x)-cos^4(x)

\int cos (x) - cos^{4} (x) d x

sin (x) - cos^{3} (x) sin (x) - \frac{3}{8} x + \frac{3}{32} sin (4 x) + C

Solution steps

\int cos (x) - cos^{4} (x) 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 cos (x) d x - \int cos^{4} (x) d x

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

\int cos^{4} (x) d x = cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x))

= sin (x) - (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)))

Simplify

sin (x) - (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x))) : sin (x) - cos^{3} (x) sin (x) - \frac{3}{8} x + \frac{3}{32} sin (4 x)

= sin (x) - cos^{3} (x) sin (x) - \frac{3}{8} x + \frac{3}{32} sin (4 x)

Add a constant to the solution

= sin (x) - cos^{3} (x) sin (x) - \frac{3}{8} x + \frac{3}{32} sin (4 x) + C

\int 9 x^{7} e^{- x^{4}} d x

y^{^{''}} - y^{^{'}} + 4 y = 2 sin (2 x)

\int_{0}^{7} 49 + x^{2} d x

\frac{d}{d x} ((x - 2) (3 x + 8))

\int \frac{x + 2}{5 x + 6} d x

What is the integral of cos(x)-cos^4(x) ?
The integral of cos(x)-cos^4(x) is sin(x)-cos^3(x)sin(x)-3/8 x+3/32 sin(4x)+C