   Chapter 7.4, Problem 40E

Chapter
Section
Textbook Problem

# Make a substitution to express the integrand as a rational function and then evaluate the integral. ∫ d x 2 x + 3 + x

To determine

To find: Make a substitution to express the integrand as a rational function and then evaluate the integral dx2x+3+x

Explanation

Calculation: Taking u(x)=x+3

We have that dudx(x)=12x+3=12u

Hence dx=2udu

and then since u23=x

We have that dx2x+3+x=2uu2+2u3du=2u(u1)(u+3)du

Now we are going to find real constants A and B such that

Au1+Bu+3=2uu2+2u3

Then, A(u+3)+B(u1)u2+2u3=2uu2+2u3 Thus we want

Taking u=1 we have that 4A=2. Hence A=12

Taking u=3 we have that 4B=6

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