Integral Love

I space left parenthesis y subscript 1 comma y subscript 2 right parenthesis equals integral subscript 1 superscript 2 open square brackets left parenthesis y subscript 1 apostrophe right parenthesis squared plus y subscript 2 end subscript superscript 2 end superscript plus left parenthesis y subscript 2 apostrophe right parenthesis squared close square brackets d x space semicolon space y subscript 1 left parenthesis 1 right parenthesis equals 1 comma space y subscript 1 left parenthesis 2 right parenthesis equals 2 comma space y subscript 2 left parenthesis 1 right parenthesis equals 0 comma space y subscript 2 left parenthesis 2 right parenthesis equals 1.