Subject: Calculus

List Of Integrals Of Irrational Functions

When doing manual integration, one of the most difficult forms to integrate belongs to the group of irrational forms. Well, not all of them but most of them. Also, the forms of irrational functions are varied that is why we will divide the integral table of irrational functions according to the form where it is most similar to.

Most common forms of irrational functions

a. Forms containing \sqrt{a+bx}

b. Forms containing a^2\pm x^2

c. Forms containing \sqrt{x^2\pm a^2}

d. Forms containing \sqrt{a^2 - x^2}

e. Forms containing \sqrt{2ax - x^2}

Integrals to the most common forms of irrational functions

Shortcut

Forms containing \sqrt{a+bx}

  • \int x\sqrt{a+bx}dx = \frac{2}{15b^3}\left (3bx-2a\right )\left (a + bx\right )^{3/2}+ C
  • \int x^2\sqrt{a+bx}dx = \frac{2}{105b^3}\left (15b^2x^2-12abx+8a^2\right )\left (a + bx\right )^{3/2}+ C
  • \int x^n\sqrt{a+bx}dx = \frac{2x^n\left (a+bx\right )^{3/2}}{\left (2n+3\right )b} - \frac{2an}{b\left (2n+3\right )} \int x^{n-1}\sqrt{a+bx}dx
  • \int \frac{x dx}{\sqrt{a+bx}} = \frac{2}{3b^2}\left (bx-2a\right )\sqrt{a+bx} + C
  • \int \frac{x^2}{\sqrt{a+bx}}dx = \frac{2}{15b^3}\left (3b^2x^2-4abx+8a^2\right )\sqrt{a+bx} + C
  • \int \frac{x^n}{\sqrt{a+bx}}dx = \frac{2x^n\sqrt{a+bx}}{b\left (2n+1\right )} - \frac{2an}{b\left (2n+1 \right )}\int \frac{x^{n-1}}{a+bx}dx
  • \int \frac{dx}{x\sqrt{a+bx}}

case 1: a > 0

= \frac{1}{\sqrt{a}}\ln{\left |\frac{\sqrt{a+bx} - \sqrt{a}}{\sqrt{a+bx} + \sqrt{a}}\right |} + C

case 1: a < 0

= \frac{2}{\sqrt{-a}}\tan^{-1}{\sqrt{\frac{a+bx}{-a}}} + C
  • \int \frac{dx}{x^n\sqrt{a+bx}} = -\frac{\sqrt{a+bx}}{a\left (n-1\right )x^{n-1}} - \frac{b\left(2n-3\right )}{2a\left (n-1\right )}\int\frac{dx}{x^{n-1}\sqrt{a+bx}}
  • \int \frac{\sqrt{a+bx}}{x}dx = 2\sqrt{a+bx} + a\int\frac{dx}{x\sqrt{a+bx}}
  • \int \frac{\sqrt{a+bx}}{x^n}dx = -\frac{\left (a+bx\right )^{3/2}}{a\left (n-1\right )x^{n-1}} - \frac{b\left(2n-5\right )}{2a\left (n-1\right )}\int \frac{\sqrt{a+bx}}{x^{n-1}}dx

Forms containing a^2\pm x^2

  • \int \frac{dx}{a^2+u^2} = \frac{1}{a}\tan^{-a}\frac{x}{a} + C
  • \int \frac{dx}{a^2-u^2} = \frac{1}{2a}\ln{\left |\frac{x+a}{x-a}\right |} + C

if |u| < a

= \frac{1}{a}\tanh^{-1}{\frac{x}{a}}+C

if |u| > a

= \frac{1}{a}\coth^{-1}{\frac{x}{a}}+C
  • \int \frac{dx}{u^2-a^2} = \frac{1}{2a}\ln{\left |\frac{x-a}{x+a}\right |} + C

if |u| < a

= -\frac{1}{a}\tanh^{-1}{\frac{x}{a}}+C

if |u| > a

= -\frac{1}{a}\coth^{-1}{\frac{x}{a}}+C

Forms containing \sqrt{x^2\pm a^2}

  • \int \frac{dx}{\sqrt{x^2\pm a^2}} = \ln{|x + \sqrt{x^2\pm a^2}|} + C
  • \int \sqrt{x^2\pm a^2}dx = \frac{x}{a} \sqrt{x^2\pm a^2} \pm \frac{a^2}{2}\ln{|x + \sqrt{x^2\pm a^2}|} + C
  • \int x^2\sqrt{x^2\pm a^2}dx = \frac{x}{8}\left (2x^2 \pm a^2\right ) \sqrt{x^2\pm a^2} - \frac{a^4}{8}\ln{|x + \sqrt{x^2\pm a^2}|} + C
  • \int \frac{\sqrt{x^2 + a^2}}{x}dx = \sqrt{x^2+a^2} - a\ln{\left |\frac{a + \sqrt{x^2 + a^2}}{x}\right |} + C
  • \int \frac{\sqrt{x^2 - a^2}}{x}dx = \sqrt{x^2 - a^2} - a\sec^{-1}\frac{x}{a} + C
  • \int \frac{\sqrt{x^2 \pm a^2}}{x^2}dx = -\frac{\sqrt{x^2 \pm a^2}}{x} + \ln{|x + \sqrt{x^2\pm a^2}|} + C
  • \int \frac{x^2}{\sqrt{x^2 \pm a^2}}dx = \frac{x}{2}\sqrt{x^2 \pm a^2} \pm \frac{a^2}{2} \ln{|x + \sqrt{x^2\pm a^2}|} + C
  • \int \frac{1}{x\sqrt{x^2 + a^2}}dx = -\frac{1}{a} \ln{\left |\frac{a + \sqrt{x^2 + a^2}}{x}\right |} + C
  • \int \frac{1}{x\sqrt{x^2 - a^2}}dx = \frac{1}{a} \sec^{-1}{\frac{x}{a}}+ C
  • \int \frac{1}{x^2\sqrt{x^2 \pm a^2}}dx = \pm\frac{1}{a^2x} \sqrt{x^2 \pm a^2} + C
  • \int \left (x^2 \pm a^2 \right )^{3/2}dx = \frac{x}{8}\left (2x^2\pm 5a^2\right ) \sqrt{x^2 \pm a^2} + \frac{3a^4}{8}\ln{\left |x + \sqrt{x^2 \pm a^2}\right |}+ C
  • \int \frac{1}{ \left (x^2 \pm a^2 \right )^{3/2} }dx = \frac{x}{a^2 \sqrt{x^2 \pm a^2} }+ C

Forms containing \sqrt{a^2 - x^2}

  • \int \frac{1}{\sqrt{a^2 - x^2}}dx = \sin^{-1}{\frac{x}{a}}+ C
  • \int \sqrt{a^2 - x^2}dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}{\frac{x}{a}}+ C
  • \int x^2\sqrt{a^2 - x^2}dx = \frac{x}{8}\left (2x^2-a^2\right )\sqrt{a^2 - x^2} + \frac{a^4}{8}\sin^{-1}{\frac{x}{a}}+ C
  • \int \frac{\sqrt{a^2 - x^2}}{x}dx = \sqrt{a^2 - x^2} - a\ln{\left |\frac{a + \sqrt{a^2-x^2}}{x}\right |} + C
  • \int \frac{\sqrt{a^2 - x^2}}{x^2}dx = -\frac{\sqrt{a^2 - x^2}}{x} - \sin^{-1}{\frac{x}{a}} C
  • \int \frac{x^2}{\sqrt{a^2 - x^2}}dx = -\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}{\frac{x}{a}} + C
  • \int \frac{1}{x\sqrt{a^2 - x^2}}dx = -\frac{1}{a}\ln{\left |\frac{a +\sqrt{a^2-x^2}}{x}\right |} + C
  • \int \frac{1}{x^2\sqrt{a^2 - x^2}}dx = -\frac{\sqrt{a^2-x^2}}{a^2x} + C
  • \int \left (a^2 - x^2\right )^{3/2}dx = -\frac{x}{8}\left (2x^2--5a^2\right )\sqrt{a^2-x^2} + \frac{3a^4}{8}\sin^{-1}{\frac{x}{a}}+C
  • \int \frac{1}{\left (a^2 - x^2\right )^{3/2}}dx = \frac{x}{a^2\sqrt{a^2--x^2}}+C

Forms containing \sqrt{2ax - x^2}

\int \sqrt{2ax-x^2}dx = \frac{x-a}{2}\sqrt{2ax-x^2}+\frac{a^2}{2}\cos^{-1}{\left (1-\frac{x}{a}\right )}+ C

\int x\sqrt{2ax-x^2}dx = \frac{2x^2-ax-3a^2}{6}\sqrt{2ax-x^2}+\frac{a^3}{2}\cos^{-1}{\left (1-\frac{x}{a}\right )}+ C

\int \frac{\sqrt{2ax-x^2}}{x}dx = \sqrt{2ax-x^2}+a\cos^{-1}{\left (1-\frac{x}{a}\right )}+ C

\int \frac{\sqrt{2ax-x^2}}{x^2}dx = -\frac{2\sqrt{2ax-x^2}}{x}-\cos^{-1}{\left (1-\frac{x}{a}\right )}+ C

\int \frac{1}{\sqrt{2ax-x^2}}dx = \cos^{-1}{\left (1-\frac{x}{a}\right )}+ C

\int \frac{x}{\sqrt{2ax-x^2}}dx = -\sqrt{2ax-x^2}+ a\cos^{-1}{\left (1-\frac{x}{a}\right )}+ C

\int \frac{x^2}{\sqrt{2ax-x^2}}dx = -\frac{x+3a}{2}\cdot \sqrt{2ax-x^2}+ \frac{3a^2}{2}\cos^{-1}{\left (1-\frac{x}{a}\right )}+ C

\int \frac{1}{x\sqrt{2ax-x^2}}dx = -\frac{\sqrt{2ax-x^2}}{ax}+ C

\int \frac{1}{\left (2ax-x^2\right )^{3/2}}dx = \frac{x-a}{a^2\sqrt{2ax-x^2}}+ C

\int \frac{u}{\left (2ax-x^2\right )^{3/2}}dx = \frac{x}{a\sqrt{2ax-x^2}}+ C