Integrate 2sin^3(x)+3.

First we need to use trigonometric identities to convert the sin^3(x) to a single power. This is because we cannot integrate trigonometric functions that are above the power of 1. We need to use the double angle formulas: sin 2x = 2 sin x cos x and cos 2x = cos^2(x) − sin^2(x) along with additional trigonometric formula sin (a + b)= sin a cos b + sin b cos a. To create a power of 3 for the sin(x) function we will have to use a combination of the two.Sin(2x+x)= sin 2x cos x+cos 2x sin x= sin(3x)=2 sin x cos x cos x + (cos^2(x)-sin^2(x))sin x =2 sin x cos^2(x) + sin x cos^2(x)-sin^3(x). As we need to substitute the sine function we re-arrange using sin^2(x) + cos^2(x)=1. Sin(3x)=2 sin x (1-sin^2(x))+ sin x (1-sin^2(x))-sin^3(x)=2 sin x- 2 sin^3(x) + sin x -sin^3(x)- sin^3(x)=3 sin x-4 sin^3(x). We then re-arrange this result to make the sin^3(x) the subject.Sin^3x=1/4(3sinx-sinx)This can now be substituted into the integral and solved. The integrand would then be (3sinx)/2-(sin3x)/2 + 3. Sin x integrates to -cos x and sin 3x would integrate to (-cos 3x)/3. Thus the integrated function would read: (-3cos x)/2 +(cos 3x)/6 +3x +c.