Question

exercises 1.5 substitution
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question 1
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answer the following true or false.
letting ( u = 4 - x^2 ) produces:
( int_{0}^{3} \frac{-2x}{4 - x^2} dx = int_{4}^{-5} \frac{1}{u} du ).
true
false
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Explanation:

Step1: Define substitution $u=4-x^2$

$u = 4 - x^2$

Step2: Compute derivative of $u$

$\frac{du}{dx} = -2x \implies du = -2x dx$

Step3: Find new limits for $u$

When $x=0$: $u=4-0^2=4$
When $x=3$: $u=4-3^2=4-9=-5$

Step4: Rewrite integral in terms of $u$

$\int_{0}^{3} \frac{-2x}{4-x^2} dx = \int_{4}^{-5} \frac{1}{u} du$

Answer:

True