Chapter 5: Q.1TB (page 416)
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) \(\int g'(h(x))h'(x)dx=g(h(x))+C\)
(b) If \(v=u^2+1\), then \(\int \sqrt{u^2+1}du=\int \sqrt{v}dv\)
(c) If \(u=x^3\), then \(\int x\sin x^3dx=\frac{1}{3x}\int \sin udu\)
(d) \(\int_0^3 u^2du=\int_0^3(u(x))^3du\)
(e) \(\int_0^1x^2dx=\int_0^1u^2du\)
(f) \(\int_2^4xe^{x^2-1}dx=\frac{1}{2}\int_2^4e^udu\)
(g) \(\int_2^3f(u(x))u'(x)dx=\int_{u(2)}^{u(3)}f(u)du\)
(h) \(\int_0^6f(u(x))u'(x)dx=\left[\int f(u)du\right]_0^6\)
Short Answer
(a) True
(b) False
(c) False
(d) False
(e) True
(f) False
(g) True
(h) False