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

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

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

(d) ${\int }_0^3{u}^{2}du={\int }_0^3(u(x){)}^{3}du$

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

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

(g) ${\int }_2^{3}f(u(x))u^{\prime }(x)dx={\int }_{u(2)}^{u(3)}f(u)du$

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

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

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

let $h(x)=t$

$h^{\prime }(x)dx=dt$

$\int g^{\prime }(t)dt=g(t)+C$

Put $t=h(x)$

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

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

$dv=2udu$

$\int \sqrt{v}\frac{dv}{2u}\ne \int \sqrt{v}dv$

It is false statement.

Part (c) Step 1: State true or false

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

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

Part (d) Step 1: State true or false

${\int }_0^3{u}^{2}du={\int }_0^3(u(x){)}^{3}du$

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

${\int }_0^1{x}^{2}dx={\int }_0^1{u}^{2}du$

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

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

Integral is correct but limit is wrong because $x\to 2,u\to 3$ and $x\to 4,u\to 15$.

It is a false statement.

Part (g) Step 1: State true or false

${\int }_2^{3}f(u(x))u^{\prime }(x)dx={\int }_{u(2)}^{u(3)}f(u)du$

Let $u(x)=t$

$u^{\prime }(x)dx=dt$

$x\to 2,t\to u(2)$

$x\to 3,t\to u(3)$

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

It is a true statement.

Part (h) Step 1: State true or false

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

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

It is false.