9x2 6x+5

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Solution

The given function is 1 9 x 2 +6x+5 .

By addition and subtraction of 1 2 in the denominator, given function can be written as,

∫ 1 9 x 2 +6x+5 dx= ∫ 1 ( 3x ) 2 +2×3x+ 1 2 +5− 1 2 dx = ∫ 1 ( 3x+1 ) 2 + 2 2 dx (1)

Also, ∫ 1 a 2 + x 2 = 1 a tan −1 x a +c (2)

Now let, 3x+1=t 3dx=dt

Substitute values of t and dt in (1),

∫ 1 ( 3x+1 ) 2 + 2 2 dx= 1 3 ∫ dt t 2 + 2 2 = 1 3 [ 1 2 tan −1 ( t 2 ) ]+c = 1 6 tan −1 ( 3x+1 2 )+c By Using (2)

Thus, the integral of the function 1 9 x 2 +6x+5 is 1 6 tan −1 ( 3x+1 2 )+c

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