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

The integral of (x+sin(x))^3(1+cos(x)) is

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

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

\frac{x ^{4}}{4} + x^{3} sin (x) - \frac{3}{4} x^{2} cos (2 x) + \frac{3}{2} x^{2} - \frac{3 x ^{2}}{4} + x sin^{3} (x) - \frac{1}{3} cos^{3} (x) + \frac{1}{3} cos^{3} (x) + \frac{1}{4} sin^{4} (x) + C

Solution steps

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

Expand

(x + sin (x))^{3} (1 + cos (x)) : x^{3} + x^{3} cos (x) + 3 x^{2} sin (x) + 3 x^{2} sin (x) cos (x) + 3 x sin^{2} (x) + 3 x sin^{2} (x) cos (x) + sin^{3} (x) + sin^{3} (x) cos (x)

= \int x^{3} + x^{3} cos (x) + 3 x^{2} sin (x) + 3 x^{2} sin (x) cos (x) + 3 x sin^{2} (x) + 3 x sin^{2} (x) cos (x) + sin^{3} (x) + sin^{3} (x) cos (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 x^{3} d x + \int x^{3} cos (x) d x + \int 3 x^{2} sin (x) d x + \int 3 x^{2} sin (x) cos (x) d x + \int 3 x sin^{2} (x) d x + \int 3 x sin^{2} (x) cos (x) d x + \int sin^{3} (x) d x + \int sin^{3} (x) cos (x) d x

\int x^{3} d x = \frac{x ^{4}}{4}

\int x^{3} cos (x) d x = x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x)))

\int 3 x^{2} sin (x) d x = 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x)))

\int 3 x^{2} sin (x) cos (x) d x = \frac{3}{2} (- \frac{1}{2} x^{2} cos (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x)))

\int 3 x sin^{2} (x) d x = 3 (\frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)))

\int 3 x sin^{2} (x) cos (x) d x = x sin^{3} (x) + cos (x) - \frac{1}{3} cos^{3} (x)

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

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

= \frac{x ^{4}}{4} + x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) + 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) + \frac{3}{2} (- \frac{1}{2} x^{2} cos (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x))) + 3 (\frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x))) + x sin^{3} (x) + cos (x) - \frac{1}{3} cos^{3} (x) - cos (x) + \frac{cos ^{3} ( x )}{3} + \frac{sin ^{4} ( x )}{4}

Simplify

\frac{x ^{4}}{4} + x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) + 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) + \frac{3}{2} (- \frac{1}{2} x^{2} cos (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x))) + 3 (\frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x))) + x sin^{3} (x) + cos (x) - \frac{1}{3} cos^{3} (x) - cos (x) + \frac{cos ^{3} ( x )}{3} + \frac{sin ^{4} ( x )}{4} : \frac{x ^{4}}{4} + x^{3} sin (x) - \frac{3}{4} x^{2} cos (2 x) + \frac{3}{2} x^{2} - \frac{3 x ^{2}}{4} + x sin^{3} (x) - \frac{1}{3} cos^{3} (x) + \frac{1}{3} cos^{3} (x) + \frac{1}{4} sin^{4} (x)

= \frac{x ^{4}}{4} + x^{3} sin (x) - \frac{3}{4} x^{2} cos (2 x) + \frac{3}{2} x^{2} - \frac{3 x ^{2}}{4} + x sin^{3} (x) - \frac{1}{3} cos^{3} (x) + \frac{1}{3} cos^{3} (x) + \frac{1}{4} sin^{4} (x)

Add a constant to the solution

= \frac{x ^{4}}{4} + x^{3} sin (x) - \frac{3}{4} x^{2} cos (2 x) + \frac{3}{2} x^{2} - \frac{3 x ^{2}}{4} + x sin^{3} (x) - \frac{1}{3} cos^{3} (x) + \frac{1}{3} cos^{3} (x) + \frac{1}{4} sin^{4} (x) + C

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