Page:Calculus Made Easy.pdf/285

(7) ${\dfrac {1}{a+1}}(\log _{\epsilon }x)^{a+1}+C$.

(8) $\log _{\epsilon }(\log _{\epsilon }x)+C$.

(9) $2\log _{\epsilon }(x-1)+3\log _{\epsilon }(x+2)+C$.

(10) ${\frac {1}{2}}\log _{\epsilon }(x-1)+{\frac {1}{5}}\log _{\epsilon }(x-2)+{\frac {3}{10}}\log _{\epsilon }(x+3)+C$.

(11) ${\dfrac {b}{2a}}\log _{\epsilon }{\dfrac {x-a}{x+a}}+C$.

(12) $\log _{\epsilon }{\dfrac {x^{2}-1}{x^{2}+1}}+C$.

(13) ${\frac {1}{4}}\log _{\epsilon }{\dfrac {1+x}{1-x}}+{\frac {1}{2}}\arctan x+C$.

(14) ${\dfrac {1}{\sqrt {a}}}\log _{\epsilon }{\dfrac {{\sqrt {a}}-{\sqrt {a-bx^{2}}}}{x{\sqrt {a}}}}$. (Let ${\dfrac {1}{x}}=v$; then, in the result, let ${\sqrt {v^{2}-{\dfrac {b}{a}}}}=v-u$.)

You had better differentiate now the answer and work back to the given expression as a check.

Every earnest student is exhorted to manufacture more examples for himself at every stage, so as to test his powers. When integrating he can always test his answer by differentiating it, to see whether he gets back the expression from which he started.

There are lots of books which give examples for practice. It will suffice here to name two: R. G. Blaine’s The Calculus and its Applications, and F. M. Saxelby’s A Course in Practical Mathematics.

 

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