   Chapter 4.5, Problem 40E

Chapter
Section
Textbook Problem

# Evaluate the definite integral. ∫ π / 3 2 π / 3 csc 2 ( 1 2 t ) d t

To determine

To evaluate:

The given definite integral π/32π/3csc212tdt.

Explanation

1) Concept:

i) The substitution rule for definite integral:

If g'(x) is a continuous function on a,b whose f is continuous on range of u=g(x), then abfgxg'(x)dx=g(a)g(b)f(u)du. Here,g(x) is substituted as u and g(x)dx =du.

ii)

csc2xdx=-cotx+C

2) Given:

π/32π/3csc212tdt

3) Calculation:

The given integral is

π/32π/3csc212tdt

Here, use the substitution method.

Substitute 12t=u   so   t=2u. Differentiating with respect to t

dt=2du

The limits change and the new limits of integration are calculated by substituting

for t=π3, u=12π3  =π6& for t=2π3, u=<

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