Whenever you have sine and cosine functions in the integral, and the term inside terms are the same, its a good idea to check if you can use the identity;
sin^2x + cos^2x=1sin2x+cos2x=1
In this case, we can rearrange so that the expression reads;
sin^2(12x)=1-cos^2(12x)sin2(12x)=1−cos2(12x)
Now we can rewrite the integral a little bit.
int sin^3(12x)cos^2(12x)dx∫sin3(12x)cos2(12x)dx
=int sin(12x)sin^2(12x)cos^2(12x)dx=∫sin(12x)sin2(12x)cos2(12x)dx
=int sin(12x)(1-cos^2(12x))cos^2(12x)dx=∫sin(12x)(1−cos2(12x))cos2(12x)dx
At first glance, this looks messier, but now we can use substitution to make the integration easier.
Let u=cos(12x)u=cos(12x)
Using the chain rule gives us;
(du)/(dx) = -12sin(12x)dudx=−12sin(12x)
-1/12 du = sin(12x)dx−112du=sin(12x)dx
Now that we have expressions for uu and dudu, lets take another look at the intgral.
int color(green)sin(12x)color(red)((1-cos^2(12x))cos^2(12x))color(green)dx∫sin(12x)(1−cos2(12x))cos2(12x)dx
We can make the dudu substitution for the green part, and the uu substitutions in the red part.
int -1/12(1-u^2)u^2du∫−112(1−u2)u2du
We can move the constant out of the integral, and multiply u^2u2 through the parenthesis, to get the integral;
-1/12 int u^2-u^4 du−112∫u2−u4du
Using the power rule, this integral is not so bad.
-1/12 (u^3/3-u^5/5) + C−112(u33−u55)+C
Multiply the -1/12−112 through.
u^5/60-u^3/36 +Cu560−u336+C
Now we can find a common denominator.
(3u^5-5u^3)/180 +C3u5−5u3180+C
Re-substituting for uu we get;
(3cos^5(12x)-5cos^3(12x))/180+ C3cos5(12x)−5cos3(12x)180+C
