Processing Math: Done

Aus Online Mathematik Brückenkurs 2

If we use the definition of tanx and write the integral as

tanxdx=sinxcosxdx

we see that the numerator sinx is the derivative of the denominator (apart from the minus sign). Hence, the substitution u=cosx will work,

sinxcosxdx=udu=cosx=(cosx)dx=−sinxdx=−udu=−lnu+C=−lncosx+C.

Note: −lncosx+C is only a primitive function in intervals in which cosx=0.