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1 solution
Set
( 1 8 x 8 + 4 8 x 7 + 1 2 x 6 + 3 6 x 5 + 9 x 2 + 1 6 x + 2 ) e x 6 − 1 = d x d [ ( a x 3 + b x 2 + c x + d ) e x 6 − 1 ] .
= ( 3 a x 2 + 2 b x + c ) e x 6 − 1 + ( 6 a x 8 + 6 b x 7 + 6 c x 6 + 6 d x 5 ) e x 6 − 1
= ( 6 a x 8 + 6 b x 7 + 6 c x 6 + 6 d x 5 + 3 a x 2 + 2 b x + c ) e x 6 − 1 .
Equating coefficients yields
a = 3 , b = 8 , c = 2 , d = 6 .
So, the antiderivative is
( 3 x 3 + 8 x 2 + 2 x + 6 ) e x 6 − 1 + C .
Evaluating from − 1 to 1 gives the final answer:
( 3 ( 1 ) 3 + 8 ( 1 ) 2 + 2 ( 1 ) + 6 ) e 1 6 − 1 − ( 3 ( − 1 ) 3 + 8 ( − 1 ) 2 + 2 ( − 1 ) + 6 ) e ( − 1 ) 6 − 1
= ( 3 + 8 + 2 + 6 ) − ( − 3 + 8 − 2 + 6 ) = 1 0 .