Evaluate this integral.
3sin(x)+4
∫ ------------- dx
(3+4sin(x))2
The trick to this integral is to start by multiplying the numerator and denominator by sec^2(x). Then we get
Integ (3*tan(x)*sec(x)+4*sec^2(x))/(3*sec(x)+4*tan(x))^2 dx
Now utilize the substitution u = 3*sec(x)+4*tan(x); du = 3*tan(x)*sec(x)+4*sec^2(x) dx. Then:
Integ 1/u^2 du -> -1/u + C
Now just back-substitute to get
-1/(3*sec(x)+4*tan(x)) + C
One little last thing is to multiply the numerator and denominator by cos(x), to put the result back into terms of sin(x). Then we have our final answer:
-cos(x)/(3+4*sin(x)) + C
Solution