Partial fractions with quadratic in denominator

3 Express \(\frac { 7 x ^ { 2 } - 3 x + 2 } { x \left( x ^ { 2 } + 1 \right) }\) in partial fractions.

5 Express \(\frac { 4 } { x \left( x ^ { 2 } + 4 \right) }\) in partial fractions.

2 Express \(\frac { 3 x } { ( 2 - x ) \left( 4 + x ^ { 2 } \right) } \quad\) in partial fractions.

4 Express \(\frac { 1 } { ( 2 x + 1 ) \left( x ^ { 2 } + 1 \right) }\) in partial fractions.

3 Express \(\frac { 3 x + 2 } { x \left( x ^ { 2 } + 1 \right) }\) in partial fractions.

4 Express \(\frac { 4 } { x \left( x ^ { 2 } + 4 \right) }\) in partial fractions.

1 It is given that \(\mathrm { f } ( x ) = \frac { 2 a x } { ( x - 2 a ) \left( x ^ { 2 } + a ^ { 2 } \right) }\), where \(a\) is a non-zero constant. Express \(\mathrm { f } ( x )\) in partial fractions.

1 Express \(\frac { 2 x + 3 } { ( x + 3 ) \left( x ^ { 2 } + 9 \right) }\) in partial fractions.

1 Express \(\frac { 3 x + 2 } { x \left( x ^ { 2 } + 1 \right) }\) in partial fractions.

1 Express \(\frac { 5 x } { ( x - 1 ) \left( x ^ { 2 } + 4 \right) }\) in partial fractions.

4 Express \(\frac { 5 x ^ { 2 } + x + 12 } { x ^ { 3 } + 4 x }\) in partial fractions.

3 Express \(\frac { x + 6 } { x \left( x ^ { 2 } + 2 \right) }\) in partial fractions.

8

1. Express \(\frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) }\) in the form \(\frac { A x + B } { x ^ { 2 } + 1 } + \frac { C } { x - 2 }\) where \(A , B\) and \(C\) are constants to be found.
2. Hence find the exact value of \(\int _ { 0 } ^ { 1 } \frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) } \mathrm { d } x\).

7 Express \(\frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in the form \(\frac { A } { x + 2 } + \frac { B x + C } { x ^ { 2 } + 1 }\) where the numerical values of \(A , B\) and \(C\) are to be found. Hence show that \(\int _ { 0 } ^ { 1 } \frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) } \mathrm { d } x = \ln 3 - \frac { 5 } { 2 } \ln 2\).

Express \(\frac{1}{(2x + 1)(x^2 + 1)}\) in partial fractions. [5]

Express \(\frac{3x}{(2-x)(4+x^2)}\) in partial fractions. [5]

Express \(\frac{5x^2+x+12}{x^3+4x}\) in partial fractions. [5]