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Consider the integral -22e53xdx.

(a) Solve this integral by using u-substitution while keeping the limits of integration in terms of x.

(b) Solve the integral again with u-substitution, this time changing the limits of integration to be in terms of u.

Short Answer

Expert verified

(a) The value of integral by using u-substitution while keeping the limits of integration in terms of x is -22e53xdx=-13e1-e11.

(b) The value of integral again with u-substitution, this time changing the limits of integration to be in terms of u is -22e53xdx=-13e-1-e11.

Step by step solution

01

Step 1. Given Information 

Consider the integral -22e53xdx.

(a) Solve this integral by using u-substitution while keeping the limits of integration in terms of x.

(b) Solve the integral again with u-substitution, this time changing the limits of integration to be in terms of u.

02

Part (a) Step 1. Now solving this integral by using u-substitution while keeping the limits of integration in terms of x. 

Let

u=53xdudx=-3-13du=dx
03

Part (a) Step 2. This substitution changes the integral into 

-22e53xdx=-13-22eudu-22e53xdx=-13eu-22-22e53xdx=-13e53x-22-22e53xdx=-13e53×2-e53×(-2)-22e53xdx=-13e56-e5+6-22e53xdx=-13e1-e11

04

Part (b) Step 2. Now solve the integral again with u-substitution, this time changing the limits of integration to be in terms of u.

Let

u=53xdudx=-3-13du=dx

05

Part (b) Step 2. We will now write the limits of integration (x=-2 and x=2) in terms of the new variable u.

When x=-2 we have

u=5-3xu(-2)=5-3(-2)u(-2)=5+6u(-2)=11

When x=2we have

u=5-3xu(2)=5-3(2)u(2)=5-6u(2)=-1

06

Part (a) Step 3. This substitution changes the integral into 

-22e53xdx=-1311-1eudu-22e53xdx=-13eu11-1-22e53xdx=-13e-1-e11

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