Question

integration 1/3x^2+13x-10dx​

Answer 1

Given : 1/(3x² + 13x - 10) dx

To Find : Integrate

Solution:

1/(3x² + 13x - 10)

= 1/(3x² + 15x - 2x - 10)

= 1/ ( 3x (x + 5 ) - 2(x + 5))

= 1/(3x - 2)(x + 5)

1/(3x - 2)(x + 5)   = A/(3x - 2)  + B/(x + 5)

=> 1  = A(x + 5)  + B (3x - 2)

x = - 5

=> 1 =  B(-17)

=> B = -1/17

x = 2/3

=> 1 = A ( 2/3 + 5)

=> A = 3/17

1/(3x - 2)(x + 5)   =   3/17(3x - 2)  -   1/17(x + 5)

∫ *1/(3x² + 13x - 10)  dx

= (1/17) ∫3/(3x - 2) dx  - (1/17)∫1/( x + 5) dx

= ( 1/17) ln | 3x - 2|  - (1/17) ln | x + 5| + C

= (1/17) ln | (3x - 2)/(x + 5) |  + C

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