What is the integral of ((1-cos^3(x))sin(x)) ?

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

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integral of ((1-cos^3(x))sin(x))

\int ((1 - cos^{3} (x)) sin (x)) d x

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

Solution steps

\int (1 - cos^{3} (x)) sin (x) d x

Expand

(1 - cos^{3} (x)) sin (x) : sin (x) - cos^{3} (x) sin (x)

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

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

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

= - cos (x) - (- \frac{cos ^{4} ( x )}{4})

Simplify

- cos (x) - (- \frac{cos ^{4} ( x )}{4}) : - cos (x) + \frac{1}{4} cos^{4} (x)

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

Add a constant to the solution

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

\int (sin^{9} (x)) cos (x) d x

x \to 0 lim (\frac{sin ( 6 x )}{x})

\int λ e^{- λ x} d x

\int 2 sin (2 t) d t

z \to 1 lim (\frac{z - 1}{z ^{2} - 1})

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