Question

Evaluate the following
(x+7)dx/x^2+14x+55

Answer 1

makes no sense as written
better look again and copy it accurately

Answer 2

If (x+7)dx/x^2+14x+55

mean

( x + 7 ) dx / ( x ^ 2 + 14 x + 55 )

then substitute :

u = x ^ 2 + 14 x + 5

d u = ( 2 x + 14 ) dx = 2 ( x + 7 ) d x Divide both sides by 2

du / 2 = ( x + 7 ) d x

( x + 7 ) d x = d u / 2


integral of [ ( x + 7 ) dx / ( x ^ 2 + 14 x + 55 ) ] =

integral of [ ( d u / 2 ) / u ) ] =

( 1 / 2 ) integral of ( du / u ) =

( 1 / 2 ) log ( u ) + C =

( 1 / 2 ) log ( x ^ 2 + 14 x + 5 ) + C =

log [ sqrt ( x ^ 2 + 14 x + 5 ) ] + C


Remark :

log is the natural logarithm

( 1 / n ) log ( a ) = log ] ( n - th root ( a ) ]

( 1 / 2 ) log ( a ) = log [ sqrt ( a ) ]