Question

Find integration of 3x+5/(x^2 +3x-1)​

Answer 1

Answer:

Step-by-step explanation:

∫3x+5 dx /(x^2 +3x-1)​

∫3x+5 dx / x^2+3x-1/4+1/4-1

∫3x+5 dx / (x-1/2)^2 - (root 3/2 )^2

∫3x dx / (x-1/2)^2 - (root3 /2)^2 +5∫dx / (x-1/2)^2 - (root3 /2)^2

3∫x dx / (x-1/2)^2 - (root3 /2)^2 +5∫dx / (x-1/2)^2 - (root3 /2)^2

=3(∫x-1/2 dx / (x-1/2)^2 - (root3 /2)^2 +1/2∫dx / (x-1/2)^2 - (root3 /2)^2 } +5∫dx / (x-1/2)^2 - (root3 /2)^2

let (x-1/2)^2=t

2(x-1/2) dx= dt

x-1/2 dx =dt/2

therefore

∫ x-1/2 dx/ (x-1/2)^2- (root3 /2)^2

∫dt/2  /  t-3/4

1/2∫dt/t-3/4

1/2ln t-3/4

ln √ t-3/4

∫ dx / (x-1/2)^2 - (root3 /2)^2

∫dx/x^2-a^2=1/2a * ln x-a/x+a

1/2*√3/2 ln x-1/2-√3 /2 / x-1/2+√3 /2

1/√3 *ln x-1/2-√3 /2 / x-1/2+√3 /2

substitute in above

3(∫x-1/2 dx / (x-1/2)^2 - (root3 /2)^2 +1/2∫dx / (x-1/2)^2 - (root3 /2)^2 } +5∫dx / (x-1/2)^2 - (root3 /2)^2

=3{ln √ t-3/4 + 1/√3 *ln x-1/2-√3 /2 / x-1/2+√3 /2 } +5/√3 *ln x-1/2-√3 /2 / x-1/2+√3 /2