What is the integral of sin^2(x)+2sin(x)+1 ?

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

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integral of sin^2(x)+2sin(x)+1

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

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

Solution steps

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

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

\int 2 sin (x) d x = - 2 cos (x)

\int 1 d x = x

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

Simplify

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

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

Add a constant to the solution

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

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