1 - Calculus

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 $\int x\sin \left(x^3\right)dx=\frac{1}{3x}\int \sin u du$

(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: ${\int }_2^4 x{e}^{x^2-1}dx=\frac{1}{2}{\int }_2^4 e^{u}du$ 

(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: 

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

Step-by-Step Solution

Answer

(part a)

(part b)

(part c)

(part d)

(part e)

(part f)

(part g)

(part h)

Step1: Introduction (part a).

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Step2: Given Information (part a).

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Step3: Explanation (part a).

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Step4: Given Information (part b).

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Step5: Explanation (part b).

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Step6: Given Information (part c).

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Step7: Explanation (part c).

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Step8: Given Information (part d).

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Step9: Explanation (part d).

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Step10: Given Information (part e).

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Step11: Explanation (part e).

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Step12: Given Information (part f).

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Step13: Explanation (part f).

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Step14: Given Information (part g).

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Step15: Explanation (part g).

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Step16: Given Information (part h).

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Step17: Explanation (part h).

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