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Question
Evaluate: `int_0^3 x^2 (3 - x)^(5/2) "d"x`
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Solution
Let I = `int_0^3 x^2 (3 - x)^(5/2) "d"x`
= `int_0^3(3 - x)^2[3 - (3 - x)]^(5/2) "d"x` ....`[∵ int_0^"a" "f"(x) "d"x = int_0^"a" "f"("a" - x)"d"x]`
= `int_0^3(9 - 6x + x^2)x^(5/2) "d"x`
= `int_0^3(9x^(5/2) - 6x^(7/2) + x^(9/2)) "d"x`
= `9int_0^2 x^(5/2) "d"x - 6 int_0^3 x^(7/2)"d"x + int_0^3 x^(9/2) "d"x`
= `9[(x^(7/2))/(7/2)]_0^3 - 6[(x^(9/2))/(9/2)]_0^3 + [(x^(11/2))/(11/2)]_0^3`
= `18/7[(3)^(7/2) - 0] - 12/9[(3)^(9/2) - 0] + 2/1[(3)^(11/2) - 0]`
= `[18/7 - (12/9 xx3) + (2/11 xx 9)](3)^(7/2)`
= `((198 - 308 + 126)/77)(3)^(7/2)`
∴ I = `16/77(3)^(7/2)`
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