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Solve the following three problems using appropriate integration techniques (substitution, integration by parts, trigonometric/substitution as needed):
β«
0
π
/
2
π₯
cos
β‘
(
π₯
)
β
π
π₯
β«
0
Ο/2
β
xcos(x)dx
β«
ln
β‘
(
π₯
)
π₯
2
β
π
π₯
β«
x
2
ln(x)
β
dx
A related-rates application problem: βA ladder 10β―ft long is sliding down a wall. At the moment the base is 6β―ft from the wall and moving outward at 2β―ft/s, how fast is the top sliding down?β
Provide full step-by-step solutions with clear reasoning.
Include evaluations of any integrals and explanations of the related-rates setup and final rates.
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