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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) โˆซgโ€ฒ(h(x))hโ€ฒ(x)dx=g(h(x))+C

(b) If v=u2+1, then โˆซu2+1du=โˆซvdv

(c) If u=x3, then โˆซxsinโกx3dx=13xโˆซsinโกudu

(d) โˆซ03u2du=โˆซ03(u(x))3du

(e) โˆซ01x2dx=โˆซ01u2du

(f) โˆซ24xex2โˆ’1dx=12โˆซ24eudu

(g) โˆซ23f(u(x))uโ€ฒ(x)dx=โˆซu(2)u(3)f(u)du

(h) โˆซ06f(u(x))uโ€ฒ(x)dx=[โˆซf(u)du]06

Short Answer

Expert verified

(a) True

(b) False

(c) False

(d) False

(e) True

(f) False

(g) True

(h) False

Step by step solution

01

Part (a) Step 1: State true or false

โˆซgโ€ฒ(h(x))hโ€ฒ(x)dx=g(h(x))+C

let h(x)=t

hโ€ฒ(x)dx=dt

โˆซgโ€ฒ(t)dt=g(t)+C

Put t=h(x)

โˆซgโ€ฒ(h(x))hโ€ฒ(x)dx=g(h(x))+C

It is a true statement.

02

Part (b) Step 1: State true or false

If v=u2+1, then โˆซu2+1du=โˆซvdv

v=u2+1

Derivative

dv=2udu

โˆซvdv2uโ‰ โˆซvdv

It is false statement.

03

Part (c) Step 1: State true or false

If u=x3, then โˆซxsinโกx3dx=13xโˆซsinโกudu

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

04

Part (d) Step 1: State true or false

โˆซ03u2du=โˆซ03(u(x))3du

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

05

Part (e) Step 1: State true or false

โˆซ01x2dx=โˆซ01u2du

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

06

Part (f) Step 1: State True or false

โˆซ24xex2โˆ’1dx=12โˆซ24eudu

Integral is correct but limit is wrong because xโ†’2,uโ†’3 and xโ†’4,uโ†’15.

It is a false statement.

07

Part (g) Step 1: State true or false

โˆซ23f(u(x))uโ€ฒ(x)dx=โˆซu(2)u(3)f(u)du

Let u(x)=t

uโ€ฒ(x)dx=dt

xโ†’2,tโ†’u(2)

xโ†’3,tโ†’u(3)

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

It is a true statement.

08

Part (h) Step 1: State true or false

โˆซ06f(u(x))uโ€ฒ(x)dx=[โˆซf(u)du]06

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

It is false.

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