Question

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

Step-by-step solution

Part (a) Step 1: State true or false

\(\int g'(h(x))h'(x)dx=g(h(x))+C\)

let \(h(x)=t\)

\(h'(x)dx=dt\)

\(\int g'(t)dt=g(t)+C\)

Put \(t=h(x)\)

\(\int g'(h(x))h'(x)dx=g(h(x))+C\)

It is a true statement.

Part (b) Step 1: State true or false

If \(v=u^2+1\), then \(\int \sqrt{u^2+1}du=\int \sqrt{v}dv\)

\(v=u^2+1\)

Derivative

\(dv=2udu\)

\(\int \sqrt{v}\frac{dv}{2u}\neq \int \sqrt{v}dv\)

It is false statement.

Part (c) Step 1: State true or false

If \(u=x^3\), then \(\int x\sin x^3dx=\frac{1}{3x}\int \sin udu\)

This is a false statement because can't take the variable \(x\) outside the integral.

Part (d) Step 1: State true or false

\(\int_0^3 u^2du=\int_0^3(u(x))^3du\)

It is false because it is integrand with variable \(x\) but integral with \(u\). It doesn't make sense.

Part (e) Step 1: State true or false

\(\int_0^1x^2dx=\int_0^1u^2du\)

This is a true statement because the limit is the same and only the variable is different. The solution will same.

Part (f) Step 1: State True or false

\(\int_2^4xe^{x^2-1}dx=\frac{1}{2}\int_2^4e^udu\)

Integral is correct but limit is wrong because \(x\rightarrow 2,u\rightarrow 3\) and \(x\rightarrow 4,u\rightarrow 15\).

It is a false statement.

Part (g) Step 1: State true or false

\(\int_2^3f(u(x))u'(x)dx=\int_{u(2)}^{u(3)}f(u)du\)

Let \(u(x)=t\)

\(u'(x)dx=dt\)

\(x\rightarrow 2,t\rightarrow u(2)\)

\(x\rightarrow 3,t\rightarrow u(3)\)

So, the limit is correct, and the integral is the same.

It is a true statement.

Part (h) Step 1: State true or false

\(\int_0^6f(u(x))u'(x)dx=\left[\int f(u)du\right]_0^6\)

Integral isn't solved and limit is applied. This is the wrong statement.

It is false.