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Question
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Solution
Let I = `int x^5sqrt(a^2 + x^2).dx`
Put, a2 + x2 = t
∴ 2x dx = dt
∴ x dx = `(1)/(2)dt`
Also, x2 = t – a2
I = `int x^2. x^2sqrt(a^2 + x^2)x dx`
=` int(t - a^2)^2 sqrt(t). dt`
= `(1)/(2) int (t^2 - 2a^2t + a^4)sqrt(t). dt`
= `(1)/(2) int (t^(5/2) - 2a^2t^(3/2) + a^4t^(1/2))dt`
= `(1)/(2) int t^(5/2) dt - a^2 int t^(3/2) dt + a^4/2 int t^(1/2) dt`
= `(1)/(2). (t^(7/2))/((7/2)) - a^2. (t^(5/2))/((5/2)) + a^4/2.(t^(3/2))/((3/2) )+ c`
= `(1)/(7)(a^2 + x^2)^(7/2) - (2a^2)/(5)(a^2 + x^2)^(5/2) + a^4/(3)(a^2 + x^2)^(3/2) + c`
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