True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True/False: $\int g^{'} (h (x)) h^{'} (x) d x = g (h (x)) + C$
(b) True/False: If $v = u^{2} + 1,$ then $\int \sqrt{u^{2} + 1} d u = \int \sqrt{v} d v$
(c) True/False: If $u = x^{3},$ then $\int x \sin (x^{3}) d x = \frac{1}{3 x} \int \sin u d u$
(d) True/False: $\int_{0}^{3} u^{2} d u = \int_{x = 0}^{x = 3} (u (x))^{2} d u$
(e) True/False: $\int_{0}^{1} x^{2} d x = \int_{0}^{1} u^{2} d u$
(f) True/False: localid="1654067255916" $\int_{2}^{4} x e^{x^{2} - 1} d x = \frac{1}{2} \int_{2}^{4} e^{u} d u$
(g) True/False: $\int_{2}^{3} f (u (x)) u^{'} (x) d x = \int_{u (2)}^{u (3)} f (u) d u$
(h) True/False:
$\int_{0}^{6} f (u (x)) u^{'} (x) d x = {[\int f (u) d u]}_{0}^{6}$

Question

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True/False: $\int g^{'} (h (x)) h^{'} (x) d x = g (h (x)) + C$
(b) True/False: If $v = u^{2} + 1,$ then $\int \sqrt{u^{2} + 1} d u = \int \sqrt{v} d v$
(c) True/False: If $u = x^{3},$ then $\int x \sin (x^{3}) d x = \frac{1}{3 x} \int \sin u d u$
(d) True/False: $\int_{0}^{3} u^{2} d u = \int_{x = 0}^{x = 3} (u (x))^{2} d u$
(e) True/False: $\int_{0}^{1} x^{2} d x = \int_{0}^{1} u^{2} d u$
(f) True/False: localid="1654067255916" $\int_{2}^{4} x e^{x^{2} - 1} d x = \frac{1}{2} \int_{2}^{4} e^{u} d u$
(g) True/False: $\int_{2}^{3} f (u (x)) u^{'} (x) d x = \int_{u (2)}^{u (3)} f (u) d u$
(h) True/False:
$\int_{0}^{6} f (u (x)) u^{'} (x) d x = {[\int f (u) d u]}_{0}^{6}$

Accepted Answer

(part a)

(part b)

(part c)

(part d)

(part e)

(part f)

(part g)

(part h)

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Step1: Introduction (part a).

Step2: Given Information (part a).

Step3: Explanation (part a).

Step4: Given Information (part b).

Step5: Explanation (part b).

Step6: Given Information (part c).

Step7: Explanation (part c).

Step8: Given Information (part d).

Step9: Explanation (part d).

Step10: Given Information (part e).

Step11: Explanation (part e).

Step12: Given Information (part f).

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Step14: Given Information (part g).

Step15: Explanation (part g).

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Step4: Given Information (part b).

Step5: Explanation (part b).

Step6: Given Information (part c).

Step7: Explanation (part c).

Step8: Given Information (part d).

Step9: Explanation (part d).

Step10: Given Information (part e).

Step11: Explanation (part e).

Step12: Given Information (part f).

Step13: Explanation (part f).

Step14: Given Information (part g).

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