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202

Calculus Made Easy

${\begin{alignedat}{4}&\log _{10}x;&&&&\int \log _{10}x\,dx&&=0.4343x(\log _{\epsilon }x-1)+C.\\&a^{x}&&{\text{(p.149)}};\quad \quad &&\int a^{x}\,dx&&={\dfrac {a^{x}}{\log _{\epsilon }a}}+C.\\&\cos ax;&&&&\int \cos ax\,dx&&={\frac {1}{a}}\sin ax+C\\\end{alignedat}}$

(for if $y=\sin ax$ , ${\dfrac {dy}{dx}}=a\cos ax$ ; hence to get $\cos ax$ one must differentiate $y={\dfrac {1}{a}}\sin ax$ ).

$\sin ax;\;\quad \quad \quad \int \sin ax\,dx=-{\frac {1}{a}}\cos ax+C.$

Try also $\cos ^{2}\theta$ ; a little dodge will simplify matters:

$\cos 2\theta =\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1$ ;

hence

$\cos ^{2}\theta ={\tfrac {1}{2}}(\cos 2\theta +1$ ,

and

${\begin{aligned}\int \cos ^{2}\theta \,d\theta &={\tfrac {1}{2}}\int (\cos 2\theta +1)\,d\theta \\&={\tfrac {1}{2}}\int \cos 2\theta \,d\theta +{\tfrac {1}{2}}\int d\theta .\\&={\frac {\sin 2\theta }{4}}+{\frac {\theta }{2}}+C.\quad {\text{(See also p. 227.)}}\end{aligned}}$

See also the Table of Standard Forms on pp. 252, 253. You should make such a table for yourself, putting in it only the general functions which you have successfully differentiated and integrated. See to it that it grows steadily!