Evaluate `int (3x+2) ^5 dx`

Answer

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Explanation

Consider the points `P(x,y),A(0,-4), and B(3,0)`. Now,
`sqrt(x^2+y^2+8y+16)-sqrt(x^2+y^2=6x+9)=5`
or `sqrt(x^2+(y+4)^(2))-sqrt ((x-3)^2+y^(2))=5` (1)
or `AP-BP=AB`
Thus, the locus of P is an infinite ray.
`sqrt(x^2+y^2+8y+16)-sqrt(x^2+y^2-6x+9)=+-5`
Therefore, the locus of point P is two rays in opposite direction.
`sqrt(x^2+y^2+8y+16)+sqrt(x^2+y^2-6x+9)=7`
or `AP+BP=AB`
Therefore, the locus of point P is line segment AB.
`sqrt(x^2+y^2+8y+16)-sqrt(x^2+y^2-6x+9)=7`
or `AP-BP=7 gt AB`
Which is not possible. Hence no locus.