Finding the derivative of an integral g(x)=int_(2x)^(6x) (u+2)/(u-4)du

Chasity Kane

Chasity Kane

Answered question

2022-10-08

Finding the derivative of an integral g ( x ) = 2 x 6 x u + 2 u 4 d u

Answer & Explanation

Kaleb Harrell

Kaleb Harrell

Beginner2022-10-09Added 14 answers

Let f ( u ) = u + 2 u 4 , and let F(u) be the antiderivative of f(u). Then
g ( x ) = d d x 2 x 6 x f ( u ) d u = d d x ( F ( u ) | 2 x 6 x ) = d d x [ F ( 6 x ) F ( 2 x ) ] = 6 F ( 6 x ) 2 F ( 2 x )
But F ( u ) = f ( u ). So the above evaluates to
6 f ( 6 x ) 2 f ( 2 x ) = 6 6 x + 2 6 x 4 2 2 x + 2 2 x 4 = .
In general,
d d x a ( x ) b ( x ) f ( u ) d u = f ( b ( x ) ) b ( x ) f ( a ( x ) ) a ( x ) .

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