What is the integral of sec^2 (3x)/1+4tan (3x)

Question

Please help me answer this, thank you!

Answer 1

This is a “u substitution” question, which you can recognize by seeing that the numerator of the integrand is basically the derivative of the denominator (times a constant).

Let u = 1 + 4tan(3x).

Then differentiate: du = 12sec²(3x)dx

Then substitute: 1/12 ∫(1/u)du

And integrate: = 1/12⋅ln|u| + C

Finally, re-substitute: 1/12⋅ln|1+4tan(3x)| + C