Integration by Substitution

1.

Consider the integral f 5 3:3 (1:4 l 1) (13:. In the following, we will evaluate the integral using two methods. A. First, rewrite the integral by multiplying out the integrand: f5$3 ($4 + 1)da: = [ jdx Then evaluate the resulting integral termbyterm: [5233(x4 + 1)dm 2C B. Next, rewrite the integral using the substitution w : $4 + 1: [59:3 (m4 + 1)dm = f jdw Evaluate this integral (and backsubstitute for w) to find the value of the original integral: f5x3(m4+l)dm=C c. How are your expressions from parts (A) and (B) different? What is the difference between the two? (answer from B)7(answer from A) = D Are both of the answers correct? Find the following indefinite integral (antiderivative). e 4x 9+ e4x dxCalculate the following integral: x 12 In(-8x) da We let u = and du So du = and v and then use the integration-by-parts formula to find that x12 In(-8x) dx = and finally, we finish evaluating to get that / x 2 In(-8x) dx =\fA mineral deposit along a strip of length 2 cm has density s(x) = 0.09x(2 - x) g/cm for 0