2(ax+b),dx= 1atan(ax+b)+C for a 0 on intervals where cos(ax+b) 0. This reverses (d)/(dx)tan(ax+b)=asec2(ax+b). Recognise the squared secant; use tangent of the same argument; divide by the coefficient of x and add C. 4sec2(3-2x),dx=-2tan(3-2x)+C. Exam check: the power rule for xn is not a rule for the exponent on a trigonometric function. Check every definite-integral interval for a zero of cosine.

\(\int\sec^2(ax+b)\,\mathrm{d}x=\frac1a\tan(ax+b)+C\) for \(a\ne0\) on intervals where \(\cos(ax+b)\ne0\).

This reverses \(\frac{\mathrm{d}}{\mathrm{d}x}\tan(ax+b)=a\sec^2(ax+b)\).

Recognise the squared secant; use tangent of the same argument; divide by the coefficient of \(x\) and add \(C\).

\(\int4\sec^2(3-2x)\,\mathrm{d}x=-2\tan(3-2x)+C\).

Exam check: the power rule for \(x^n\) is not a rule for the exponent on a trigonometric function. Check every definite-integral interval for a zero of cosine.

Integrating secant squared

Integrating secant squared