Consider the integral ∫ 4x(x^2 + 1) dx. In the following, we will evaluate the integral using two methods. Remember to write the arbitrary constant as "C".

Question

Consider the integral . In the following, we will evaluate the integral using two methods. Remember to write the arbitrary constant as "C".

A. First, rewrite the integral by multiplying out the integrand:

 4x³ + 4x 

Then evaluate the resulting integral term-by-term:

x⁴ + 2x² + C

B. Next, rewrite the integral using the substitution w = x² + 1:

 ? 

Evaluate this integral (and back-substitute for w) to find the value of the original integral:

 x⁴ + 2x² + C

C. How are your expressions from parts (A) and (B) different? What is the difference between the two? (Ignore the constant of integration.)

(answer from B) - (answer from A) =

Answer 1

The substitution gives you x=sqrt(w-1) and dx=1/[2sqrt(w-1) dw.

Then the integrand becomes 2w, which integrates to w².

Now back substitute and compare your 2 expressions.