Question

Show that ∫ 7x-x^2 dx / (2-x)(1+x^2) = 7/2 ln 2-π/4

Answer 1

I=∫{(7x-x²)/[(2-x)(1+x²)]}dx. (No limits given for a definite integral)

Express in partial fractions:

Let (7x-x²)/[(2-x)(1+x²)]=A/(2-x)+(Bx+C)/(1+x²);

7x-x²=A+Ax²+(Bx+C)(2-x)=A+Ax²+2Bx-Bx²+2C-Cx.

Equating coefficients:

constant: A+2C=0, C=-A/2,

x: 2B-C=7, 2B+A/2=7 (1)

x²: A-B=-1, 2A-2B=-2 (2)

(1)+(2)=5A/2=5, A=2⇒B=3, C=-1.

I=2∫dx/(2-x)+3∫(x/(1+x²))dx-∫dx/(1+x²),

I=-2ln(2-x)+3/2ln(1+x²)-tan^-1(x).

If the limits were [0,1], then I=3/2ln(2)-π/4.