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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

\int 1 d x = x

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

Simplify

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

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

Add a constant to the solution

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

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