009B Sample Midterm 2, Problem 3

Evaluate

a) $\int _{1}^{2}{\bigg (}2t+{\frac {3}{t^{2}}}{\bigg )}{\bigg (}4t^{2}-{\frac {5}{t}}{\bigg )}dt$

b) $\int _{0}^{2}(x^{3}+x){\sqrt {x^{4}+2x^{2}+4}}dx$

Foundations:
Integrating polynomials
U substitution

Solution:

(a)

Step 1:
We multiply the product inside the integral to get
$\int _{1}^{2}{\bigg (}2t+{\frac {3}{t^{2}}}{\bigg )}{\bigg (}4t^{2}-{\frac {5}{t}}{\bigg )}dt=\int _{1}^{2}{\bigg (}8t^{3}-10+12-{\frac {15}{t^{3}}}{\bigg )}dt=\int _{1}^{2}(8t^{3}+2-15t^{-3})dt$

Step 2:
We integrate to get
$\int _{1}^{2}{\bigg (}2t+{\frac {3}{t^{2}}}{\bigg )}{\bigg (}4t^{2}-{\frac {5}{t}}{\bigg )}dt=\left.2t^{4}+2t+{\frac {15}{2}}t^{-2}\right|_{1}^{2}$.
We now evaluate to get
$\int _{1}^{2}{\bigg (}2t+{\frac {3}{t^{2}}}{\bigg )}{\bigg (}4t^{2}-{\frac {5}{t}}{\bigg )}dt=2(2)^{4}+2(2)+{\frac {15}{2(2)^{2}}}-{\bigg (}2+2+{\frac {15}{2}}{\bigg )}=36+{\frac {15}{8}}-4-{\frac {15}{2}}={\frac {211}{8}}$

(b)

Step 1:

Step 2:

Final Answer:
(a) ${\frac {211}{8}}$

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