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

The integral of (1+cos(x))^3 is 5/2 x+3/4 sin(2x)+4sin(x)-1/3 sin^3(x)+C

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

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

\frac{5}{2} x + \frac{3}{4} sin (2 x) + 4 sin (x) - \frac{1}{3} sin^{3} (x) + C

Solution steps

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

Expand

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

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

\int 1 d x = x

\int 3 cos (x) d x = 3 sin (x)

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

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

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

Simplify

x + 3 sin (x) + \frac{3}{2} (x + \frac{1}{2} sin (2 x)) + sin (x) - \frac{sin ^{3} ( x )}{3} : \frac{5}{2} x + \frac{3}{4} sin (2 x) + 4 sin (x) - \frac{1}{3} sin^{3} (x)

= \frac{5}{2} x + \frac{3}{4} sin (2 x) + 4 sin (x) - \frac{1}{3} sin^{3} (x)

Add a constant to the solution

= \frac{5}{2} x + \frac{3}{4} sin (2 x) + 4 sin (x) - \frac{1}{3} sin^{3} (x) + C

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What is the integral of (1+cos(x))^3 ?
The integral of (1+cos(x))^3 is 5/2 x+3/4 sin(2x)+4sin(x)-1/3 sin^3(x)+C