Maths > Indefinite Integrals > 8.0 Integration using partial fractions
Indefinite Integrals
1.0 Introduction
2.0 Methods of Integration
3.0 Integration by parts
4.0 Integral of the type $\int {{e^x}\left\{ {f(x) + f'(x)} \right\}dx} $
5.0 Integral of the type $\int {\frac{{dx}}{{a{x^2} + bx + c}},\int {\frac{{dx}}{{\sqrt {a{x^2} + bx + c} }},\int {\sqrt {a{x^2} + bx + c} } dx} } $
6.0 Integral of the type $\int {\frac{{px + q}}{{a{x^2} + bx + c}}dx,} \int {\frac{{px + q}}{{\sqrt {a{x^2} + bx + c} }}dx,} \int {(px + q)} \sqrt {a{x^2} + bx + c} dx$
7.0 Integral of the type $\int {\frac{{a{x^2} + bx + c}}{{(p{x^2} + qx + r)}}dx,\int {\frac{{a{x^2} + bx + c}}{{\sqrt {p{x^2} + qx + r} }}dx} } ,\int {(a{x^2} + bx + c)} \sqrt {p{x^2} + qx + r} dx$
8.0 Integration using partial fractions
8.1 Type A: Linear and non-repeating
8.2 Type B: Linear and repeating
8.3 Type C: Quadratic and non-repeating
8.4 Type D: Quadratic and repeating
9.0 Integration of trigonometric functions
10.0 Integral of type $\int {({{\sin }^m}x}$${\cos ^n}x)dx$
11. Integral of type $\int {\frac{{{x^2} \pm 1}}{{{x^4} + k{x^2} + 1}}dx} $
12. Integration of irrational algebraic functions
13.0 Integral of type $\int {{x^m}{{\left( {a + b{x^n}} \right)}^p}dx} $
14.0 Reduction formulae
8.4 Type D: Quadratic and repeating
8.2 Type B: Linear and repeating
8.3 Type C: Quadratic and non-repeating
8.4 Type D: Quadratic and repeating
Let us assume $$g(x) = {(a{x^2} + bx + c)^k}(x - {d_1})(x - {d_2})...(x - {d_r})$$
Therefore, rational algebraic function is written as $$\frac{{f(x)}}{{g(x)}} = \frac{{{A_1}x + {A_2}}}{{a{x^2} + bx + c}} + \frac{{{A_3}x + {A_4}}}{{{{\left( {a{x^2} + bx + c} \right)}^2}}} + ... + \frac{{{A_{2k - 1}}x + {A_{2k}}}}{{{{\left( {a{x^2} + bx + c} \right)}^k}}} + \frac{{{D_1}}}{{x - {d_1}}} + \frac{{{D_2}}}{{x - {d_2}}} + ... + \frac{{{D_r}}}{{x - {d_r}}}$$
Case II: When degree of $f(x)$ is greater than degree of $g(x)$ then $\frac{{f(x)}}{{g(x)}}$ is called improper rational function which is expressed in the form of $$\phi (x) + \frac{{\chi (x)}}{{g(x)}}$$ where $\phi (x)$ and ${\chi (x)}$ are polynomials such that the degree of ${\chi (x)}$ becomes less than degree of $g(x)$.
After this the integral can be solved using the above mentioned Type A, B, C and D.
