Answer:
= 1/2 tan^(-1)(2 cot(x)) + constant
Step-by-step explanation:
Take the integral:
integral1/(3 sin^2(x) - 4) dx
Multiply numerator and denominator of 1/(3 sin^2(x) - 4) by -sec^2(x):
= integral-(sec^2(x))/(4 sec^2(x) - 3 tan^2(x)) dx
Prepare to substitute u = tan(x). Rewrite -(sec^2(x))/(4 sec^2(x) - 3 tan^2(x)) using sec^2(x) = tan^2(x) + 1:
= integral-(sec^2(x))/(tan^2(x) + 4) dx
For the integrand -(sec^2(x))/(tan^2(x) + 4), substitute u = tan(x) and du = sec^2(x) dx:
= integral-1/(u^2 + 4) du
Factor out constants:
= - integral1/(u^2 + 4) du
Factor 4 from the denominator:
= - integral1/(4 (u^2/4 + 1)) du
Factor out constants:
= -1/4 integral1/(u^2/4 + 1) du
For the integrand 1/(u^2/4 + 1), substitute s = u/2 and ds = 1/2 du:
= -1/2 integral1/(s^2 + 1) ds
The integral of 1/(s^2 + 1) is tan^(-1)(s):
= -1/2 tan^(-1)(s) + constant
Substitute back for s = u/2:
= -1/2 tan^(-1)(u/2) + constant
Substitute back for u = tan(x):
= -1/2 tan^(-1)(tan(x)/2) + constant
Which is equivalent for restricted x values to:
Answer: = 1/2 tan^(-1)(2 cot(x)) + constant
