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Chapter 4 | Applications of Derivatives
Evaluate each of the following indefinite integrals: a. ∫ ⎛ ⎝ 5 x 3 −7 x 2 +3 x +4 ⎞ ⎠ dx b. ∫ x 2 +4 x 3 x dx c. ∫ 4 1+ x 2 dx d. ∫ tan x cos xdx
Solution a. Using Properties of Indefinite Integrals , we can integrate each of the four terms in the integrand separately. We obtain ∫ ⎛ ⎝ 5 x 3 −7 x 2 +3 x +4 ⎞ ⎠ dx = ∫ 5 x 3 dx − ∫ 7 x 2 dx + ∫ 3 xdx + ∫ 4 dx . From the second part of Properties of Indefinite Integrals , each coefficient can be written in front of the integral sign, which gives ∫ 5 x 3 dx − ∫ 7 x 2 dx + ∫ 3 xdx + ∫ 4 dx =5 ∫ x 3 dx −7 ∫ x 2 dx +3 ∫ xdx +4 ∫ 1 dx . Using the power rule for integrals, we conclude that ∫ ⎛ ⎝ 5 x 3 −7 x 2 +3 x +4 ⎞ ⎠ dx = 5 4 x 4 − 7 3 x 3 + 3 2 x 2 +4 x + C . b. Rewrite the integrand as x 2 +4 x 3 x = x 2 x + 4 x 3 x =0. Then, to evaluate the integral, integrate each of these terms separately. Using the power rule, we have ∫ ⎛ ⎝ x + 4 x 2/3 ⎞ ⎠ dx = ∫ xdx +4 ∫ x −2/3 dx = 1 2 x 2 +4 1 ⎛ ⎝ −2 3 ⎞ ⎠ +1 x (−2/3)+1 + C = 1 2 x 2 +12 x 1/3 + C . c. Using Properties of Indefinite Integrals , write the integral as 4 ∫ 1 1+ x 2 dx . Then, use the fact that tan −1 ( x ) is an antiderivative of 1 ⎛ ⎝ 1+ x 2 ⎞ ⎠ to conclude that
∫ 4
dx =4tan −1 ( x )+ C .
1+ x 2
d. Rewrite the integrand as
tan x cos x = sin x
cos x cos x = sin x .
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