Simplify the integral, if necessary, to reduce it to an inverse form. For example, the integral ∫ tan (x) dx, which does not appear to be an inverse, simplifies using "u - substitution" to ∫ tan (x) dx = ∫ ( sin (x) / cos (x) )dx. Substituting "u" for cosine of x, ∫ ( sin (x) / cos (x) )dx = ∫ - (1 / u) du. This final form is the negative inverse of "u."
Solve the integral by converting to the natural log of the denominator in the integrand. For example, ∫ - (1 / u) du yields, - ln | u | + C. Notice the absolute value sign and the newly added "+C." The added term is the "constant of integration." As a student, it is vital to remember to add this term in all integrals which are indefinite, that is, lacking limits of integration.
Back substitute the original value of "u" into the final answer. In conclusion, Cosine of x was substituted, therefore substituting it back yields -ln | cos (x) | + C. This may be further simplified, if desired, to ln | sec (x) | + C.