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Question
Evaluate the following definite integrals:
`int_0^(pi/2) sqrt(cos theta) sin^3theta "d"theta`
Solution
I = `int_0^(pi/2) sqrt(cos theta) sin^3theta "d"theta`
= `int_0^(pi/2) sqrt(cos theta) (1 - cos^2theta)sintheta "d"theta`
t = cos θ
dt = sinθ dθ
| θ | 0 | `pi/2` |
| t | 1 | 0 |
= `int_1^0 sqrt("t") (1 - "t"^2)(- "dt")`
= `int_0^1 (sqrt("t") - "t"^2 sqrt("t")) "dt"`
= `int_0^1 ("t"^(1/2) - "t"^(5/2)) "dt"`
= `["t"^(1/2)/(3/2) - "t"^(7/2)/(7/2)]_0^1`
= `2/3 - 2/7`
= `(14 - 6)/21`
= `8/21`
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