1.

Evaluate the following integral : int(x+sqrt(a^(2)+x^(2)))^(2007)dx

Answer»

Solution :Let us substitute `x+SQRT(a^(2)+x^(2))=t` so that `sqrt(a^(2)+x^(2)-x=(a^(2))/(t)`
`implies2x=t-(a^(2))/(t)`
`impliesdx=(1)/(2)(1+(a^(2))/(t^(2)))DT`
under given substitution, the given integral reduces to
`int(x+sqrt(a^(2)+x^(2)))^(2007)dx=intt^(2007)*(1)/(2)(1+(a^(2))/(t^(2)))dt`
`=(1)/(2)int(t^(2007)+a^(2)t^(2005))dt`
`=(1)/(2)[(t^(2008))/(2008)+a^(2)*(t^(2006))/(2006)]+k`
`=(1)/(2)(x+sqrt(a^(2)+x^(2)))^(2006)[((x+sqrt(a^(2)+x^(2)))^(2))/(2008)+(a^(2))/(2006)]+k`


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