calculus
posted by ZUkii .
Evaluate the following
(x+7)dx/x^2+14x+55

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