Integrals, Trig Substitutions with quartics: &#x222B;<!-- ∫ --> <mrow class="MJX-TeXAtom-ORD"

Justine Webster

Justine Webster

Answered question

2022-05-14

Integrals, Trig Substitutions with quartics: x ( x 2 + 2 x + 5 ) 2 d x

Answer & Explanation

Jerry Kidd

Jerry Kidd

Beginner2022-05-15Added 18 answers

Hint. By observing that
( x 2 + 2 x + 5 ) 2 = [ ( x + 1 ) 2 + 4 ] 2
the change of variable
x + 1 = 2 tan u , d x = 2 ( tan 2 u + 1 ) d u ,
gives
x ( x 2 + 2 x + 5 ) 2 d x = 1 8 ( 2 cos u sin u cos 2 u ) d u
which might be easier to evaluate.

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