InterviewSolution
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InterviewSolution
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Evaluate `intxsqrt(x+x^(2))dx`. |
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Answer» Let `x=A*(d)/(dx)(x+x^(2))+B`. Then, x=A(1+2x)+B . . . (i) Comparing the coefficients of various powers of x , we get `(2A=1andA+B=0)rArr(A=(1)/(2)andB=(-1)/(2))`. `thereforex=(1)/(2)(1+2x)-(1)/(2)` `rArrintxsqrt(x+x^(2))dx` `=int{(1)/(2)(1+2x)-(1)/(2)}sqrt(x+x^(2))dx` `=(1)/(2)int(1+2x)sqrt(x+x^(2))dx-(1)/(2)intsqrt(x+x^(2))dx` `=(1)/(2)intsqrt(t)dt-(1)/(2)intsqrt((x^(2)+x+(1)/(4))-(1)/(4))}dx`, where ` (x+x^(2))` = t in the first integral `=(1)/(2)*(t^(3//2))/((3//2))-(1)/(2)intsqrt({(x+(1)/(2))^(2)-((1)/(2))^(2)})dx` `=(1)/(3)(x+x^(2))^(3//2)-(1)/(2)*intsqrt(u^(2)-((1)/(2))^(2))du` , where `(x+(1)/(2))=u` `=(1)/(3)(x+x^(2))^(3//2)-(1)/(2){(u)/(2)*sqrt(u^(2)(-1)/(4))-(1)/(8)log|u+sqrt(u^(2)-(1)/(4))|}+C` `{becauseintsqrt(x^(2)-a^(2))dx=(x)/(2)sqrt(x^(2)-a^(2))-(a^(2))/(2)log|x+sqrt(x^(2)-a^(2))|+C}` `=(1)/(3)(x+x^(2))^(3//2)-(1)/(4)(x+(1)/(2))sqrt((x+(1)/(2))^(2)-(1)/(4))+(1)/(16)log|(x+(1)/(2))sqrt((x+(1)/(2))^(2)-(1)/(4))|+C` `=(1)/(3)(x+x^(2))^(3//2)-(1)/(8)(2x+1)sqrt(x+x^(2))+(1)/(16)log|((2x+1))/(2)*sqrt(x+x^(2))|+C`. |
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