Question

1 2x 3Jo5x2 +114.

Answer 1

Consider the given integral,

I= ∫ 0 1 2x+3 5 x 2 +1 dx

Now, integrate the function,

∫ 2x+3 5 x 2 +1 dx = ∫ 2x 5 x 2 +1 dx + ∫ 3 5 x 2 +1 dx = 1 5 ∫ 10x 5 x 2 +1 dx +3 ∫ 1 5 x 2 +1 dx = 1 5 ∫ 10x 5 x 2 +1 dx +3 ∫ 1 5( x 2 + 1 5 ) dx )

Further simplify,

∫ 2x+3 5 x 2 +1 dx = 1 5 log( 5 x 2 +1 )+ 3 5 1 1 5 tan −1 x 1 5 = 1 5 log( 5 x 2 +1 )+ 3 5 tan −1 ( 5 x ) =F( x )

By fundamental theorem of calculus, we get

I=F( 1 )−F( 0 ) =[ 1 5 log( 5+1 )+ 3 5 tan −1 5 ]−[ 1 5 log( 0+1 )+ 3 5 tan −1 ( 0 ) ] = 1 5 log6+ 3 5 tan −1 5

Thus, the solution of integral is 1 5 log6+ 3 5 tan −1 5 .