Not a very difficult metamorphosis!

There are whole treatises, such as Boole’s Differential Equations, devoted to the subject of thus finding the “solutions” for different original forms.

Exercises XIX (See p. 264 for Answers.)

(1) Find ${\displaystyle \int {\sqrt {a^{2}-x^{2}}}\,dx}$.

(2) Find ${\displaystyle \int x\log _{\epsilon }x\,dx}$.

(3) Find ${\displaystyle \int x^{a}\log _{\epsilon }x\,dx}$.

(4) Find ${\displaystyle \int \epsilon ^{x}\cos \epsilon ^{x}\,dx}$.

(5) Find ${\displaystyle \int {\dfrac {1}{x}}\cos(\log _{\epsilon }x)\,dx}$.

(6) Find ${\displaystyle \int x^{2}\epsilon ^{x}\,dx}$.

(7) Find ${\displaystyle \int {\dfrac {(\log _{\epsilon }x)^{a}}{x}}\,dx}$.

(8) Find ${\displaystyle \int {\dfrac {dx}{x\log _{\epsilon }x}}}$.

(9) Find ${\displaystyle \int {\dfrac {5x+1}{x^{2}+x-2}}\,dx}$.

(10) Find ${\displaystyle \int {\dfrac {(x^{2}-3)\,dx}{x^{3}-7x+6}}}$.

(11) Find ${\displaystyle \int {\dfrac {b\,dx}{x^{2}-a^{2}}}}$.

(12) Find ${\displaystyle \int {\dfrac {4x\,dx}{x^{4}-1}}}$.

(13) Find ${\displaystyle \int {\dfrac {dx}{1-x^{4}}}}$.

(14) Find ${\displaystyle \int {\dfrac {dx}{x{\sqrt {a-bx^{2}}}}}}$.