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Calculus

# Integrals of Irrational Functions

Argument (independent variable): $$x$$

Real numbers: $$C$$, $$a$$, $$b$$, $$c$$

1. An algebraic function involving one or more radicals of polynomials is called an irrational function. Integrals of irrational functions usually contain linear, quadratic or linear fractional expressions under the radical sign.
2. $${\large\int {\frac{{dx}}{{\sqrt {ax + b} }}}\normalsize} =$$ $${\large\frac{2}{a}\normalsize}\sqrt {ax + b} + C$$
3. $${\large\int\normalsize} {\sqrt {ax + b}\,dx} =$$ $${\large\frac{2}{{3a}}\normalsize}{\left( {ax + b} \right)^{3/2}} + C$$
4. $${\large\int {\frac{{xdx}}{{\sqrt {ax + b} }}}\normalsize} =$$ $${\large\frac{{2\left( {ax – 2b} \right)}}{{3{a^2}}}\normalsize}\sqrt {ax + b} + C$$
5. $${\large\int\normalsize} {x\sqrt {ax + b}\,dx} =$$ $${\large\frac{{2\left( {3ax – 2b} \right)}}{{15{a^2}}}\normalsize}{\left( {ax + b} \right)^{3/2}}$$ $$+\; C$$
6. $${\large\int {\frac{{dx}}{{\left( {x + c} \right)\sqrt {ax + b} }}}\normalsize} =$$ $${\large\frac{1}{{\sqrt {b – ac} }}\normalsize} \ln \left| {\large\frac{{\sqrt {ax + b} – \sqrt {b – ac} }}{{\sqrt {ax + b} + \sqrt {b – ac} }}\normalsize} \right|$$ $$+\; C,$$ $$b – ac \gt 0.$$
7. $${\large\int {\frac{{dx}}{{\left( {x + c} \right)\sqrt {ax + b} }}}\normalsize} =$$ $${\large\frac{1}{{\sqrt {ac – b} }}\normalsize} \arctan\sqrt {\large\frac{{ax + b}}{{ac – b}}\normalsize}$$ $$+\; C,$$ $$b – ac \lt 0.$$
8. $${\large\int\normalsize} {\sqrt {{\large\frac{{ax + b}}{{cx + d}}}\normalsize} \,dx} =$$ $${\large\frac{1}{c}\normalsize}\sqrt {\left( {ax + b} \right)\left( {cx + d} \right)}$$ $$-\; {\large\frac{{ad – bc}}{{c\sqrt {ac} }}\normalsize}\ln \big| {\sqrt {a\left( {cx + d} \right)} }$$ $$+\;{ \sqrt {c\left( {ax + b} \right)} } \big|$$ $$+\; C,$$ $$a \gt 0.$$
9. $${\large\int\normalsize} {\sqrt {{\large\frac{{ax + b}}{{cx + d}}}\normalsize} \,dx} =$$ $${\large\frac{1}{c}\normalsize}\sqrt {\left( {ax + b} \right)\left( {cx + d} \right)}$$ $$-\; {\large\frac{{ad – bc}}{{c\sqrt {ac} }}\normalsize} \arctan\sqrt {\large\frac{{a\left( {cx + d} \right)}}{{c\left( {ax + b} \right)}}\normalsize}$$ $$+\; C,$$ $$a \lt 0,c \gt 0.$$
10. $${\large\int\normalsize} {{x^2}\sqrt {ax + b}\,dx} =$$ $${\large\frac{{2\left( {8{a^2} – 12abx + 15{b^2}{x^2}} \right)}}{{105{b^3}}}\normalsize}$$ $$\sqrt {{{\left( {ax + b} \right)}^3}}$$ $$+\;C$$
11. $${\large\int {\frac{{{x^2}dx}}{{\sqrt {ax + b} }}}\normalsize} =$$ $${\large\frac{{2\left( {8{a^2} – 4abx + 3{b^2}{x^2}} \right)}}{{15{b^3}}}\normalsize}$$ $$\sqrt {ax + b}$$ $$+\; C$$
12. $${\large\int {\frac{{dx}}{{x\sqrt {a + bx} }}}\normalsize} =$$ $${\large\frac{1}{{\sqrt a }}\normalsize}\ln \left| {\large\frac{{\sqrt {a + bx} – \sqrt a }}{{\sqrt {a + bx} + \sqrt a }}\normalsize} \right|$$ $$+\; C,$$ $$a \gt 0.$$
13. $${\large\int {\frac{{dx}}{{x\sqrt {a + bx} }}}\normalsize} =$$ $${\large\frac{2}{{\sqrt { – a} }}\normalsize} \arctan\left| {\large\frac{{a + bx}}{{ – a}}\normalsize} \right| + C,$$ $$a \lt 0.$$
14. $${\large\int\normalsize} {\sqrt {{\large\frac{{a – x}}{{b + x}}}\,dx}\normalsize} =$$ $$\sqrt {\left( {a – x} \right)\left( {b + x} \right)}$$ $$+\; \left( {a + b} \right)\arcsin \sqrt {\large\frac{{x + b}}{{a + b}}\normalsize}$$ $$+\; C$$
15. $${\large\int\normalsize} {\sqrt {{\large\frac{{a + x}}{{b – x}}}\,dx}\normalsize} =$$ $$-\sqrt {\left( {a + x} \right)\left( {b – x} \right)}$$ $$-\; \left( {a + b} \right)\arcsin \sqrt {\large\frac{{b – x}}{{a + b}}\normalsize}$$ $$+\; C$$
16. $${\large\int\normalsize} {\sqrt {{\large\frac{{1 + x}}{{1 – x}}}\normalsize}\,dx} =$$ $$– \sqrt {1 – {x^2}}$$ $$+\; \arcsin x + C$$
17. $${\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {\left( {x – a} \right)\left( {b – a} \right)} }}\normalsize}} =$$ $${ 2\arcsin \sqrt {\large\frac{{x – a}}{{b – a}}} }\normalsize + C$$
18. $${\large\int\normalsize} {\sqrt {a + bx – c{x^2}}\,dx} =$$ $${\large\frac{{2cx – b}}{{4c}}\normalsize}\sqrt {a + bx – c{x^2}}$$ $$+\;{\large\frac{{{b^2} – 4ac}}{{8\sqrt {{c^3}} }}\normalsize}\arcsin {\large\frac{{2cx – b}}{{\sqrt {{b^2} + 4ac} }}\normalsize}$$ $$+\; C$$
19. $${\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {a{x^2} + bx + c} }}}\normalsize} =$$ $${\large\frac{1}{{\sqrt a }}\normalsize}\ln \big| {2ax + b }$$ $$+{\; 2\sqrt {a\left( {a{x^2} + bx + c} \right)} } \big|$$ $$+\; C,$$ $$a \gt 0.$$
20. $${\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {a{x^2} + bx + c} }}}\normalsize} =$$ $$-{\large\frac{1}{{\sqrt { – a} }}\normalsize} \arcsin{\large\frac{{2ax + b}}{{4a}}\normalsize}$$ $$\sqrt {{b^2} – 4ac}$$ $$+\; C,$$ $$a \lt 0.$$
21. $${\large\int\normalsize} {\sqrt {{x^2} + {a^2}} dx} =$$ $${\large\frac{x}{2}\normalsize}\sqrt {{x^2} + {a^2}}$$ $$+\;{\large\frac{{{a^2}}}{2}\normalsize}\ln\left| {x + \sqrt {{x^2} + {a^2}} } \right|$$ $$+\; C$$
22. $${\large\int\normalsize} {x\sqrt {{x^2} + {a^2}} dx} =$$ $${\large\frac{1}{3}\normalsize}{\left( {{x^2} + {a^2}} \right)^{3/2}} + C$$
23. $${\large\int\normalsize} {{x^2}\sqrt {{x^2} + {a^2}} dx} =$$ $${\large\frac{x}{8}\normalsize}\left( {2{x^2} + {a^2}} \right)\sqrt {{x^2} + {a^2}}$$ $$-\;{\large\frac{{{a^4}}}{8}\normalsize}\ln \left| {x + \sqrt {{x^2} + {a^2}} } \right|$$ $$+\; C$$
24. $${\large\int\normalsize} {{\large\frac{{\sqrt {{x^2} + {a^2}} }}{{{x^2}}}\normalsize} dx} =$$ $$– {\large\frac{{\sqrt {{x^2} + {a^2}} }}{x}\normalsize}$$ $$+\; \ln \left| {x + \sqrt {{x^2} + {a^2}} } \right| + C$$
25. $${\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {{x^2} + {a^2}} }}}\normalsize} =$$ $$\ln \left| {x + \sqrt {{x^2} + {a^2}} } \right| + C$$
26. $${\large\int\normalsize} {{\large\frac{{\sqrt {{x^2} + {a^2}} }}{x}\normalsize}\,dx} =$$ $$\sqrt {{x^2} + {a^2}}$$ $$+\; a\ln \left| {\large\frac{x}{{a + \sqrt {{x^2} + {a^2}} }}\normalsize} \right| + C$$
27. $${\large\int\normalsize} {\large{\frac{{xdx}}{{\sqrt {{x^2} + {a^2}} }}}\normalsize} =$$ $$\sqrt {{x^2} + {a^2}} + C$$
28. $${\large\int\normalsize} {\large{\frac{{{x^2}dx}}{{\sqrt {{x^2} + {a^2}} }}}\normalsize} =$$ $${\large\frac{x}{2}\normalsize}\sqrt {{x^2} + {a^2}}$$ $$-\; {\large\frac{{{a^2}}}{2}\normalsize}\ln \left| {x + \sqrt {{x^2} + {a^2}} } \right|$$ $$+\; C$$
29. $${\large\int\normalsize} {\large{\frac{{dx}}{{x\sqrt {{x^2} + {a^2}} }}}\normalsize} =$$ $${\large\frac{1}{a}\normalsize}\ln \left| {\large\frac{x}{{a + \sqrt {{x^2} + {a^2}} }}\normalsize} \right| + C$$
30. $${\large\int\normalsize} {\sqrt {{x^2} – {a^2}} dx} =$$ $${\large\frac{x}{2}\normalsize}\sqrt {{x^2} – {a^2}}$$ $$-\;{\large\frac{{{a^2}}}{2}\normalsize}\ln \left| {x + \sqrt {{x^2} – {a^2}} } \right|$$ $$+\; C$$
31. $${\large\int\normalsize} {x\sqrt {{x^2} – {a^2}} dx} =$$ $${\large\frac{1}{3}\normalsize}{\left( {{x^2} – {a^2}} \right)^{3/2}} + C$$
32. $${\large\int\normalsize} {{\large\frac{{\sqrt {{x^2} – {a^2}} }}{x}\normalsize} dx} =$$ $$\sqrt {{x^2} – {a^2}}$$ $$+\; a\arcsin {\large\frac{a}{x}\normalsize} + C$$
33. $${\large\int\normalsize} {{\large\frac{{\sqrt {{x^2} – {a^2}} }}{{{x^2}}}\normalsize} dx} =$$ $$-{\large\frac{{\sqrt {{x^2} – {a^2}} }}{x}\normalsize}$$ $$+\; \ln \left| {x + \sqrt {{x^2} – {a^2}} } \right|$$ $$+\; C$$
34. $${\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {{x^2} – {a^2}} }}}\normalsize} =$$ $$\ln \left| {x + \sqrt {{x^2} – {a^2}} } \right| + C$$
35. $${\large\int\normalsize} {\large{\frac{{xdx}}{{\sqrt {{x^2} – {a^2}} }}}\normalsize} =$$ $$\sqrt {{x^2} – {a^2}} + C$$
36. $${\large\int\normalsize} {\large{\frac{{{x^2}dx}}{{\sqrt {{x^2} – {a^2}} }}}\normalsize} =$$ $${\large\frac{x}{2}\normalsize}\sqrt {{x^2} – {a^2}}$$ $$+\;{\large\frac{{{a^2}}}{2}\normalsize}\ln \left| {x + \sqrt {{x^2} – {a^2}} } \right|$$ $$+\; C$$
37. $${\large\int\normalsize} {\large{\frac{{dx}}{{x\sqrt {{x^2} – {a^2}} }}}\normalsize} =$$ $$– {\large\frac{1}{a}\normalsize}\arcsin {\large\frac{a}{x}\normalsize} + C$$
38. $${\large\int\normalsize} {\large{\frac{{dx}}{{\left( {x + a} \right)\sqrt {{x^2} – {a^2}} }}}\normalsize} =$$ $${\large\frac{1}{a}\normalsize}\sqrt {\large\frac{{x – a}}{{x + a}}\normalsize} + C$$
39. $${\large\int\normalsize} {\large{\frac{{dx}}{{\left( {x – a} \right)\sqrt {{x^2} – {a^2}} }}}\normalsize} =$$ $$-{\large\frac{1}{a}\normalsize}\sqrt {\large\frac{{x + a}}{{x – a}}\normalsize} + C$$
40. $${\large\int\normalsize} {\large{\frac{{dx}}{{{x^2}\sqrt {{x^2} – {a^2}} }}}\normalsize} =$$ $${\large\frac{{\sqrt {{x^2} – {a^2}} }}{{{a^2}x}}\normalsize} + C$$
41. $${\large\int\normalsize} {\large{\frac{{dx}}{{{{\left( {{x^2} – {a^2}} \right)}^{3/2}}}}}\normalsize} =$$ $$– {\large\frac{x}{{{a^2}\sqrt {{x^2} – {a^2}} }}\normalsize} + C$$
42. $${\large\int\normalsize} {{{\left( {{x^2} – {a^2}} \right)}^{3/2}}dx} =$$ $$-\;{\large\frac{x}{8}\normalsize}\left( {2{x^2} – 5{a^2}} \right)\sqrt {{x^2} – {a^2}}$$ $$+\;{\large\frac{{3{a^4}}}{8}\normalsize}\ln \left| {x + \sqrt {{x^2} – {a^2}} } \right|$$ $$+\; C$$
43. $${\large\int\normalsize} {\sqrt {{a^2} – {x^2}} dx} =$$ $${\large\frac{x}{2}\normalsize}\sqrt {{a^2} – {x^2}}$$ $$+\;{\large\frac{{{a^2}}}{2}\normalsize}\arcsin {\large\frac{x}{a}\normalsize} + C$$
44. $${\large\int\normalsize} {x\sqrt {{a^2} – {x^2}} dx} =$$ $$– {\large\frac{1}{3}\normalsize}{\left( {{a^2} – {x^2}} \right)^{3/2}} + C$$
45. $${\large\int\normalsize} {{x^2}\sqrt {{a^2} – {x^2}} dx} =$$ $${\large\frac{x}{8}\normalsize}\left( {2{x^2} – {a^2}} \right)\sqrt {{a^2} – {x^2}}$$ $$+\;{\large\frac{{{a^4}}}{8}\normalsize}\arcsin {\large\frac{x}{a}\normalsize} + C$$
46. $${\large\int\normalsize} {{\large\frac{{\sqrt {{a^2} – {x^2}} }}{x}\normalsize} dx} =$$ $$\sqrt {{a^2} – {x^2}}$$ $$+\; a\ln \left| {\large\frac{x}{{a + \sqrt {{a^2} – {x^2}} }}\normalsize} \right| + C$$
47. $${\large\int\normalsize} {{\large\frac{{\sqrt {{a^2} – {x^2}} }}{{{x^2}}}\normalsize} dx} =$$ $$– {\large\frac{{\sqrt {{a^2} – {x^2}} }}{x}\normalsize}$$ $$-\; \arcsin {\large\frac{x}{a}\normalsize} + C$$
48. $${\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {1 – {x^2}} }}}\normalsize} =$$ $$\arcsin x + C$$
49. $${\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {{a^2} – {x^2}} }}}\normalsize} =$$ $$\arcsin {\large\frac{x}{a}\normalsize} + C$$
50. $${\large\int\normalsize} {\large{\frac{{xdx}}{{\sqrt {{a^2} – {x^2}} }}}\normalsize} =$$ $$– \sqrt {{a^2} – {x^2}} + C$$
51. $${\large\int\normalsize} {\large{\frac{{{x^2}dx}}{{\sqrt {{a^2} – {x^2}} }}}\normalsize} =$$ $$– {\large\frac{x}{2}\normalsize}\sqrt {{a^2} – {x^2}}$$ $$+\;{\large\frac{{{a^2}}}{2}\normalsize}\arcsin {\large\frac{x}{a}\normalsize} + C$$
52. $${\large\int\normalsize} {\large{\frac{{dx}}{{\left( {x + a} \right)\sqrt {{a^2} – {x^2}} }}}\normalsize} =$$ $$– {\large\frac{1}{2}\normalsize}\sqrt {\large\frac{{a – x}}{{a + x}}\normalsize} + C$$
53. $${\large\int\normalsize} {\large{\frac{{dx}}{{\left( {x – a} \right)\sqrt {{a^2} – {x^2}} }}}\normalsize} =$$ $$– {\large\frac{1}{2}\normalsize}\sqrt {\large\frac{{a + x}}{{a – x}}\normalsize} + C$$
54. $${\large\int\normalsize} {\large\frac{{dx}}{{\left( {x + b} \right)\sqrt {{a^2} – {x^2}} }}\normalsize} =$$ $${\large\frac{1}{{\sqrt {{b^2} – {a^2}} }}\normalsize}\arcsin {\large\frac{{bx + {a^2}}}{{a\left( {x + b} \right)}}\normalsize}$$ $$+\; C,$$ $$b \gt a.$$
55. $${\large\int\normalsize} {\large{\frac{{dx}}{{\left( {x + b} \right)\sqrt {{a^2} – {x^2}} }}}\normalsize} =$$ $${\large\frac{1}{{\sqrt {{a^2} – {b^2}} }}\normalsize}$$ $$\ln\left| {\large\frac{{x + b}}{{\sqrt {{a^2} – {b^2}} \sqrt {{a^2} – {x^2}} + {a^2} + bx}}\normalsize} \right|$$ $$+\;C,$$ $$b \gt a.$$
56. $${\large\int\normalsize} {\large{\frac{{dx}}{{{x^2}\sqrt {{a^2} – {x^2}} }}}\normalsize} =$$ $$– {\large\frac{{\sqrt {{a^2} – {x^2}} }}{{{a^2}x}}\normalsize} + C$$
57. $${\large\int\normalsize} {{{\left( {{a^2} – {x^2}} \right)}^{3/2}}dx} =$$ $${\large\frac{x}{8}\normalsize}\left( {5{a^2} – 2{x^2}} \right)\sqrt {{a^2} – {x^2}}$$ $$+\;{\large\frac{{3{a^4}}}{8}\normalsize}\arcsin {\large\frac{x}{a}\normalsize} + C$$
58. $${\large\int\normalsize} {\large{\frac{{dx}}{{{{\left( {{a^2} – {x^2}} \right)}^{3/2}}}}}\normalsize} =$$ $${\large\frac{x}{{{a^2}\sqrt {{a^2} – {x^2}} }}\normalsize} + C$$