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: $\int g^{\mathrm{\prime }}\left(h\right(x\left)\right)h^{\mathrm{\prime }}\left(x\right)dx=g\left(h\right(x\left)\right)+C$

(b) True/False: If $v=u^2+1,$then $\int \sqrt{u^2+1}du=\int \sqrt{v}dv$

(c) True/False: If $u=x^3,$then $\int x\sin \left(x^3\right)dx=\frac{1}{3x}\int \sin udu$

(d) True/False: ${\int }_0^3 u^{2}du={\int }_{x=0}^{x=3} \left(u\right(x){)}^{2}du$

(e) True/False:${\int }_0^1 x^{2}dx={\int }_0^1 u^{2}du$

(f) True/False: localid="1654067255916" ${\int }_2^4 x{e}^{x^2-1}dx=\frac{1}{2}{\int }_2^4 e^{u}du$

(g) True/False: ${\int }_2^3 f\left(u\right(x\left)\right)u^{\mathrm{\prime }}\left(x\right)dx={\int }_{u\left(2\right)}^{u\left(3\right)} f\left(u\right)du$

(h) True/False:

${\int }_0^6 f\left(u\right(x\left)\right)u^{\mathrm{\prime }}\left(x\right)dx={\left[\int f\left(u\right)du\right]}_0^6$

Short answer

(part a)

(part b)

(part c)

(part d)

(part e)

(part f)

(part g)

(part h)

Step-by-step solution

Introduction (part a).

1

Given Information (part a).

2

Explanation (part a).

3

Given Information (part b).

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Explanation (part b).

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Given Information (part c).

6

Explanation (part c).

7

Given Information (part d).

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Explanation (part d).

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Given Information (part e).

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Explanation (part e).

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Given Information (part f).

12

Explanation (part f).

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Given Information (part g).

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Explanation (part g).

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Given Information (part h).

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Explanation (part h).

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